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Section 1: Analytical
Analytical Temperature Results x_bar = 0.5:
x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
k2/k1 = 0.0625 0.0 0.011667 0.025181 0.042681 0.066937 0.066937 0.847794 4.228179 20.573130 100.0
k2/k1 = 0.125 0.0 0.097641 0.210739 0.357195 0.560194 0.560194 3.303981 10.626282 32.674022 100.0
k2/k1 = 0.25 0.0 0.464798 1.003171 1.700341 2.666666 2.666666 8.626608 20.264910 45.242343 100.0
k2/k1 = 0.5 0.0 1.470842 3.174511 5.380689 8.438605 8.438605 17.437015 32.027881 56.890833 100.0
k2/k1 = 1.0 0.0 3.440413 7.425426 12.585848 19.738549 19.738549 30.015765 45.044332 67.203196 100.0
k2/k1 = 2.0 0.0 6.351689 13.708819 23.235988 36.441301 36.441301 45.815586 58.792577 76.392720 100.0
k2/k1 = 4.0 0.0 9.653341 20.834759 35.314219 55.383747 55.383747 62.464754 71.993744 84.344156 100.0
k2/k1 = 8.0 0.0 12.567937 27.125318 45.976504 72.105546 72.105546 76.701212 82.797389 90.513336 100.0
k2/k1 = 16.0 0.0 14.652807 31.625083 53.603455 84.066993 84.066993 86.741990 90.264767 94.669752 100.0
Analytical Temperature Results x_bar = 2/pi:
x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.637 x = 0.750 x = 0.875 x = 1.000
k2/k1 = 0.0625 0.0 0.043265 0.093379 0.158275 0.248224 0.377466 0.392138 4.086217 20.545109 100.0
k2/k1 = 0.125 0.0 0.217770 0.470012 0.796654 1.249403 1.899926 1.973771 10.004385 32.490342 100.0
k2/k1 = 0.25 0.0 0.723845 1.562272 2.647999 4.152891 6.315166 6.560619 18.929061 44.739811 100.0
k2/k1 = 0.5 0.0 1.775153 3.831304 6.493931 10.184517 15.487261 16.089210 30.677503 56.308955 100.0
k2/k1 = 1.0 0.0 3.440413 7.425426 12.585848 19.738549 30.015765 31.182398 45.044332 67.203196 100.0
k2/k1 = 2.0 0.0 5.484175 11.836467 20.062415 31.464146 47.846498 49.706164 60.645947 77.284348 100.0
k2/k1 = 4.0 0.0 7.434698 16.046273 27.197894 42.654810 64.863775 67.384857 74.763593 85.702464 100.0
k2/k1 = 8.0 0.0 8.921798 19.255874 32.638061 51.186694 77.837932 80.863285 85.279652 91.742445 100.0
k2/k1 = 16.0 0.0 9.877475 21.318505 36.134152 56.669658 86.175696 89.525116 91.966805 95.516633 100.0
Analytical Heat Convection Results x_bar = 0.5:
Heat Convection
k2/k1 = 0.0625 -1.241829
k2/k1 = 0.125 -1.756437
k2/k1 = 0.25 -2.486946
k2/k1 = 0.5 -3.529489
k2/k1 = 1.0 -4.985127
k2/k1 = 2.0 -6.832006
k2/k1 = 4.0 -8.797049
k2/k1 = 8.0 -10.486214
k2/k1 = 16.0 -11.680625
Analytical Heat Convection Results x_bar = 2/pi:
Heat Convection
k2/k1 = 0.0625 -1.241978
k2/k1 = 0.125 -1.758799
k2/k1 = 0.25 -2.501202
k2/k1 = 0.5 -3.564213
k2/k1 = 1.0 -4.985127
k2/k1 = 2.0 -6.610807
k2/k1 = 4.0 -8.118713
k2/k1 = 8.0 -9.254592
k2/k1 = 16.0 -9.980639
Section 2: FDM with Varying k_ratio
FDM 2nd Order Temperature Results x_bar = 0.5:
x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
k2/k1 = 0.0625 0.0 0.023566 0.050814 0.086002 0.134628 1.249213 5.486829 23.441518 100.0
k2/k1 = 0.125 0.0 0.131160 0.282813 0.478656 0.749289 3.850963 11.766340 34.389643 100.0
k2/k1 = 0.25 0.0 0.522237 1.126073 1.905858 2.983433 9.158380 21.057315 46.117073 100.0
k2/k1 = 0.5 0.0 1.534785 3.309379 5.601064 8.767916 17.841592 32.490766 57.293304 100.0
k2/k1 = 1.0 0.0 3.487383 7.519670 12.726906 19.922720 30.231460 45.263865 67.368749 100.0
k2/k1 = 2.0 0.0 6.373779 13.743460 23.260557 36.412116 45.832593 58.833740 76.431273 100.0
k2/k1 = 4.0 0.0 9.664493 20.839062 35.269736 55.211305 62.353389 71.931152 84.318726 100.0
k2/k1 = 8.0 0.0 12.587755 27.142347 45.937931 71.911316 76.562507 82.709060 90.471024 100.0
k2/k1 = 16.0 0.0 14.689868 31.675027 53.609410 83.920263 86.634225 90.194225 94.635027 100.0
FDM 4th Order Temperature Results x_bar = 0.5:
x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
k2/k1 = 0.0625 0.0 0.012407 0.026778 0.045387 0.071181 0.872956 4.311784 20.775791 100.0
k2/k1 = 0.125 0.0 0.099063 0.213805 0.362390 0.568335 3.324588 10.669465 32.740427 100.0
k2/k1 = 0.25 0.0 0.466246 1.006291 1.705616 2.674914 8.638042 20.281132 45.260122 100.0
k2/k1 = 0.5 0.0 1.471712 3.176373 5.383805 8.443410 17.441951 32.033043 56.895148 100.0
k2/k1 = 1.0 0.0 3.440655 7.425912 12.586575 19.739499 30.016879 45.045466 67.204053 100.0
k2/k1 = 2.0 0.0 6.351568 13.708490 23.235255 36.439800 45.814649 58.792070 76.392524 100.0
k2/k1 = 4.0 0.0 9.653156 20.834256 35.313096 55.381453 62.463124 71.992703 84.343652 100.0
k2/k1 = 8.0 0.0 12.567869 27.125037 45.975677 72.103555 76.699755 82.796434 90.512864 100.0
k2/k1 = 16.0 0.0 14.652898 31.625124 53.603116 84.065651 86.740995 90.264109 94.669425 100.0
FDM 2nd Order Temperature Results x_bar = 2/pi:
x = 0.000 x = 0.159 x = 0.318 x = 0.477 x = 0.637 x = 0.727 x = 0.818 x = 0.909 x = 1.000
k2/k1 = 0.0625 0.0 0.077055 0.173629 0.314183 0.534322 3.515595 11.139042 33.471059 100.0
k2/k1 = 0.125 0.0 0.319719 0.720423 1.303613 2.217011 8.401972 20.134133 45.159372 100.0
k2/k1 = 0.25 0.0 0.980497 2.209358 3.997855 6.799021 16.282888 31.141948 56.281369 100.0
k2/k1 = 0.5 0.0 2.330072 5.250358 9.500576 16.157318 27.426136 43.221817 66.151538 100.0
k2/k1 = 1.0 0.0 4.469024 10.070065 18.221884 30.989359 41.796025 56.052044 74.933939 100.0
k2/k1 = 2.0 0.0 7.113289 16.028395 29.003540 49.325368 57.925784 68.916457 82.750907 100.0
k2/k1 = 4.0 0.0 9.660074 21.767073 39.387737 66.985431 72.825251 80.167607 89.163985 100.0
k2/k1 = 8.0 0.0 11.617645 26.178075 47.369488 80.559732 84.071349 88.450249 93.741604 100.0
k2/k1 = 16.0 0.0 12.883057 29.029430 52.529043 89.334419 91.281472 93.699357 96.600545 100.0
FDM 4th Order Temperature Results x_bar = 2/pi:
x = 0.000 x = 0.159 x = 0.318 x = 0.477 x = 0.637 x = 0.727 x = 0.818 x = 0.909 x = 1.000
k2/k1 = 0.0625 0.0 0.056906 0.128532 0.233402 0.398641 3.008594 10.028399 31.747344 100.0
k2/k1 = 0.125 0.0 0.282730 0.638587 1.159615 1.980577 7.908859 19.346064 44.258786 100.0
k2/k1 = 0.25 0.0 0.937019 2.116398 3.843183 6.564006 15.927669 30.693906 55.871337 100.0
k2/k1 = 0.5 0.0 2.296558 5.187121 9.419332 16.087851 27.280752 43.038453 65.997640 100.0
k2/k1 = 1.0 0.0 4.450576 10.052292 18.254031 31.177173 41.886144 56.075679 74.924868 100.0
k2/k1 = 2.0 0.0 7.094631 16.024287 29.098623 49.699312 58.180432 69.070572 82.820650 100.0
k2/k1 = 4.0 0.0 9.618383 21.724560 39.449790 67.378702 73.109297 80.350881 89.253120 100.0
k2/k1 = 8.0 0.0 11.542702 26.070922 47.342381 80.858940 84.291953 88.595273 93.813330 100.0
k2/k1 = 16.0 0.0 12.779433 28.864263 52.414830 89.522484 91.421372 93.792016 96.646648 100.0
FDM Heat Convection Results for x_bar = 0.5:
2nd Order Heat Convection 4th Order Heat Convection
k2/k1 = 0.0625 -1.583037 -1.603973
k2/k1 = 0.125 -2.012353 -2.038259
k2/k1 = 0.25 -2.674530 -2.701452
k2/k1 = 0.5 -3.665089 -3.690105
k2/k1 = 1.0 -5.082312 -5.103008
k2/k1 = 2.0 -6.905215 -6.914954
k2/k1 = 4.0 -8.864016 -8.851486
k2/k1 = 8.0 -10.561319 -10.519257
k2/k1 = 16.0 -11.768665 -11.699504
FDM Heat Convection Results for x_bar = 2/pi:
2nd Order Heat Convection 4th Order Heat Convection
k2/k1 = 0.0625 -1.432463 -1.253695
k2/k1 = 0.125 -1.898803 -1.763475
k2/k1 = 0.25 -2.603336 -2.502977
k2/k1 = 0.5 -3.639854 -3.564905
k2/k1 = 1.0 -5.047653 -4.985553
k2/k1 = 2.0 -6.678556 -6.611422
k2/k1 = 4.0 -8.208090 -8.119747
k2/k1 = 8.0 -9.370579 -9.256061
k2/k1 = 16.0 -10.118258 -9.982436
Section 2: FEM with Varying k_ratio
FEM p=1 Temperature Results x_bar = 0.5:
x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
k2/k1 = 0.0625 0.0 0.008040 0.017369 0.029485 0.046331 0.431694 2.667176 16.333414 100.0
k2/k1 = 0.125 0.0 0.092421 0.199668 0.338947 0.532604 2.747608 9.300941 30.539969 100.0
k2/k1 = 0.25 0.0 0.461952 0.998014 1.694185 2.662150 8.197855 19.452994 44.279989 100.0
k2/k1 = 0.5 0.0 1.463251 3.161248 5.366397 8.432466 17.199749 31.637279 56.504681 100.0
k2/k1 = 1.0 0.0 3.408346 7.363486 12.499933 19.641717 29.852520 44.852497 67.048062 100.0
k2/k1 = 2.0 0.0 6.282365 13.572595 23.040247 36.204196 45.614623 58.635706 76.298138 100.0
k2/k1 = 4.0 0.0 9.556088 20.645239 35.046460 55.070107 62.227195 71.830962 84.259013 100.0
k2/k1 = 8.0 0.0 12.463187 26.925816 45.708094 71.823222 76.490410 82.656431 90.442107 100.0
k2/k1 = 16.0 0.0 14.553527 31.441845 53.374310 83.869497 86.594484 90.166500 94.620483 100.0
FEM p=2 Temperature Results x_bar = 0.5:
x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
k2/k1 = 0.0625 0.0 0.012068 0.026047 0.044148 0.069238 0.867408 4.294089 20.733162 100.0
k2/k1 = 0.125 0.0 0.098218 0.211984 0.359304 0.563499 3.316456 10.653512 32.716155 100.0
k2/k1 = 0.25 0.0 0.465290 1.004231 1.702131 2.669460 8.632229 20.273710 45.252255 100.0
k2/k1 = 0.5 0.0 1.471126 3.175116 5.381693 8.440137 17.439069 32.030325 56.892999 100.0
k2/k1 = 1.0 0.0 3.440538 7.425677 12.586223 19.739039 30.016340 45.044917 67.203638 100.0
k2/k1 = 2.0 0.0 6.351749 13.708915 23.236059 36.441232 45.815587 58.792618 76.392764 100.0
k2/k1 = 4.0 0.0 9.653407 20.834849 35.314232 55.383492 62.464580 71.993637 84.344107 100.0
k2/k1 = 8.0 0.0 12.568039 27.125469 45.976580 72.105307 76.701038 82.797275 90.513280 100.0
k2/k1 = 16.0 0.0 14.652946 31.625303 53.603617 84.066829 86.741869 90.264686 94.669712 100.0
FEM p=1 Temperature Results x_bar = 2/pi:
x = 0.000 x = 0.159 x = 0.318 x = 0.477 x = 0.637 x = 0.727 x = 0.818 x = 0.909 x = 1.000
k2/k1 = 0.0625 0.0 0.051867 0.117450 0.214096 0.367363 2.434575 8.623646 29.412987 100.0
k2/k1 = 0.125 0.0 0.275874 0.624708 1.138757 1.953971 7.451625 18.477330 43.210596 100.0
k2/k1 = 0.25 0.0 0.917965 2.078702 3.789190 6.501798 15.648464 30.261653 55.446231 100.0
k2/k1 = 0.5 0.0 2.241155 5.075024 9.251075 15.873740 27.039252 42.794017 65.812018 100.0
k2/k1 = 1.0 0.0 4.339198 9.825975 17.911409 30.733838 41.542855 55.828147 74.785094 100.0
k2/k1 = 2.0 0.0 6.933484 15.700652 28.620142 49.108745 57.744459 68.779447 82.672214 100.0
k2/k1 = 4.0 0.0 9.434441 21.363990 38.943632 66.822621 72.697960 80.078383 89.116689 100.0
k2/k1 = 8.0 0.0 11.358900 25.721867 46.887445 80.453254 83.990156 88.394995 93.713291 100.0
k2/k1 = 16.0 0.0 12.604001 28.541357 52.026991 89.272101 91.234400 93.667692 96.584539 100.0
FEM p=2 Temperature Results x_bar = 2/pi:
x = 0.000 x = 0.159 x = 0.318 x = 0.477 x = 0.637 x = 0.727 x = 0.818 x = 0.909 x = 1.000
k2/k1 = 0.0625 0.0 0.056356 0.127290 0.231150 0.394802 3.000696 10.011857 31.721242 100.0
k2/k1 = 0.125 0.0 0.282083 0.637133 1.156990 1.976128 7.903319 19.338326 44.250198 100.0
k2/k1 = 0.25 0.0 0.936679 2.115650 3.841878 6.561891 15.925358 30.691359 55.869129 100.0
k2/k1 = 0.5 0.0 2.296689 5.187467 9.420092 16.089426 27.281478 43.038618 65.997532 100.0
k2/k1 = 1.0 0.0 4.451072 10.053508 18.256499 31.181922 41.889346 56.077632 74.925778 100.0
k2/k1 = 2.0 0.0 7.095212 16.025752 29.101694 49.705410 58.184789 69.073373 82.822018 100.0
k2/k1 = 4.0 0.0 9.618769 21.725639 39.452308 67.384157 73.113291 80.353498 89.254414 100.0
k2/k1 = 8.0 0.0 11.542780 26.071346 47.343822 80.862788 84.294805 88.597158 93.814267 100.0
k2/k1 = 16.0 0.0 12.779243 28.864110 52.415295 89.524814 91.423109 93.793169 96.647223 100.0
FEM Heat Convection Results for x_bar = 0.5:
p=1 Heat Convection p=2 Heat Convection
k2/k1 = 0.0625 -1.638864 -1.604308
k2/k1 = 0.125 -2.072823 -2.038640
k2/k1 = 0.25 -2.732243 -2.701699
k2/k1 = 0.5 -3.714639 -3.690240
k2/k1 = 1.0 -5.122610 -5.103060
k2/k1 = 2.0 -6.938675 -6.914893
k2/k1 = 4.0 -8.894031 -8.851258
k2/k1 = 8.0 -10.590389 -10.518839
k2/k1 = 16.0 -11.797909 -11.698927
FEM Heat Convection Results for x_bar = 2/pi:
p=1 Heat Convection p=2 Heat Convection
k2/k1 = 0.0625 -1.476318 -1.253926
k2/k1 = 0.125 -1.940923 -1.763642
k2/k1 = 0.25 -2.639437 -2.503068
k2/k1 = 0.5 -3.669207 -3.564914
k2/k1 = 1.0 -5.073389 -4.985398
k2/k1 = 2.0 -6.705769 -6.610952
k2/k1 = 4.0 -8.240802 -8.118855
k2/k1 = 8.0 -9.409744 -9.254767
k2/k1 = 16.0 -10.162539 -9.980847
Section 3 Temp: Convergence of FDM and FEM Temp with Varying k Ratio
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 0.0625
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 0.066937 0.400229 NaN 4.979197 NaN NaN NaN 0.104810 NaN 5.658100e-01 NaN NaN NaN
N = 8 0.066937 0.134628 NaN 1.011272 -2.299743 NaN NaN 0.071181 NaN 6.340026e-02 -3.157757 NaN NaN
N = 16 0.066937 0.082453 0.069697 0.183010 -2.125242 0.183010 NaN 0.067277 0.066765 7.677697e-03 -3.640568 7.677697e-03 NaN
N = 32 0.066937 0.070718 0.067312 0.050588 -2.036995 0.050588 -2.152561 0.066960 0.066932 4.195795e-04 -3.891466 4.195795e-04 -3.620631
N = 64 0.066937 0.067876 0.066968 0.013561 -2.009682 0.013561 -2.045906 0.066938 0.066937 2.326877e-05 -3.971390 2.326877e-05 -3.885859
N = 128 0.066937 0.067171 0.066939 0.003469 -2.002449 0.003469 -2.012080 0.066937 0.066937 1.395107e-06 -3.992773 1.395107e-06 -3.969945
N = 256 0.066937 0.066995 0.066937 0.000873 -2.000614 0.000873 -2.003060 0.066937 0.066937 8.620313e-08 -3.998578 8.620313e-08 -3.992525
N = 512 0.066937 0.066952 0.066937 0.000219 -2.000154 0.000219 -2.000768 0.066937 0.066937 5.560008e-09 -4.005552 5.560008e-09 -3.951767
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 0.0625
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 0.066937 -0.363773 NaN 6.434572 NaN NaN NaN 0.125341 NaN 8.725219e-01 NaN NaN NaN
N = 8 0.066937 0.046331 NaN 0.307843 -4.385579 NaN NaN 0.069238 NaN 3.437611e-02 -4.665713 NaN NaN
N = 16 0.066937 0.062673 0.063352 0.010706 -2.272928 0.010706 NaN 0.067069 0.066982 1.301621e-03 -4.117651 1.301621e-03 NaN
N = 32 0.066937 0.065888 0.066675 0.011806 -2.023150 0.011806 -2.345888 0.066945 0.066937 1.131235e-04 -4.025230 1.131235e-04 -4.123498
N = 64 0.066937 0.066671 0.066924 0.003772 -1.982240 0.003772 -2.036753 0.066937 0.066937 7.455960e-06 -4.006027 7.455960e-06 -4.026495
N = 128 0.066937 0.066870 0.066937 0.001005 -1.984061 0.001005 -1.981623 0.066937 0.066937 4.709251e-07 -4.001558 4.709251e-07 -4.006325
N = 256 0.066937 0.066920 0.066937 0.000254 -1.990328 0.000254 -1.981947 0.066937 0.066937 2.948335e-08 -4.001514 2.948335e-08 -4.002541
N = 512 0.066937 0.066933 0.066937 0.000064 -1.994748 0.000064 -1.988845 0.066937 0.066937 2.462299e-09 -4.018498 2.462299e-09 -3.613957
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 0.125
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 0.560194 1.321586 NaN 1.359157 NaN NaN NaN 0.644433 NaN 1.503736e-01 NaN NaN NaN
N = 8 0.560194 0.749289 NaN 0.337552 -2.009531 NaN NaN 0.568335 NaN 1.453274e-02 -3.371173 NaN NaN
N = 16 0.560194 0.607201 0.560273 0.083760 -2.008170 0.083760 NaN 0.560783 0.559951 1.485903e-03 -3.788394 1.485903e-03 NaN
N = 32 0.560194 0.571925 0.560274 0.020794 -2.002601 0.020794 -2.010016 0.560233 0.560189 7.737417e-05 -3.942014 7.737417e-05 -3.777069
N = 64 0.560194 0.563125 0.560201 0.005220 -2.000690 0.005220 -2.003237 0.560197 0.560194 4.471350e-06 -3.985145 4.471350e-06 -3.939060
N = 128 0.560194 0.560927 0.560195 0.001307 -2.000175 0.001307 -2.000862 0.560194 0.560194 2.731656e-07 -3.996264 2.731656e-07 -3.984413
N = 256 0.560194 0.560377 0.560194 0.000327 -2.000044 0.000327 -2.000219 0.560194 0.560194 1.695761e-08 -3.999156 1.695761e-08 -3.997252
N = 512 0.560194 0.560240 0.560194 0.000082 -2.000011 0.000082 -2.000055 0.560194 0.560194 1.457216e-09 -4.000608 1.457216e-09 -3.572239
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 0.125
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 0.560194 0.284007 NaN 0.493020 NaN NaN NaN 0.622268 NaN 1.108077e-01 NaN NaN NaN
N = 8 0.560194 0.532604 NaN 0.049252 -3.323401 NaN NaN 0.563499 NaN 5.898885e-03 -4.231472 NaN NaN
N = 16 0.560194 0.553343 0.555230 0.003400 -2.009667 0.003400 NaN 0.560394 0.560221 3.090697e-04 -4.046545 3.090697e-04 NaN
N = 32 0.560194 0.558340 0.559926 0.002833 -1.885500 0.002833 -2.053153 0.560207 0.560195 2.153037e-05 -4.010664 2.153037e-05 -4.048887
N = 64 0.560194 0.559704 0.560217 0.000915 -1.920215 0.000915 -1.872828 0.560195 0.560194 1.372921e-06 -4.002600 1.372921e-06 -4.011199
N = 128 0.560194 0.560068 0.560200 0.000236 -1.955459 0.000236 -1.907766 0.560194 0.560194 8.618886e-08 -4.000648 8.618886e-08 -4.002640
N = 256 0.560194 0.560162 0.560195 0.000059 -1.976687 0.000059 -1.948157 0.560194 0.560194 5.269850e-09 -4.000264 5.269850e-09 -4.031720
N = 512 0.560194 0.560186 0.560194 0.000015 -1.988098 0.000015 -1.972821 0.560194 0.560194 1.244402e-09 -4.007095 1.244402e-09 -1.615813
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 0.25
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 2.666666 3.862069 NaN 0.448276 NaN NaN NaN 2.766445 NaN 3.741715e-02 NaN NaN NaN
N = 8 2.666666 2.983433 NaN 0.118788 -1.916002 NaN NaN 2.674914 NaN 3.092841e-03 -3.596695 NaN NaN
N = 16 2.666666 2.747136 2.660209 0.032677 -1.976907 0.032677 NaN 2.667225 2.666520 2.643938e-04 -3.882410 2.643938e-04 NaN
N = 32 2.666666 2.686866 2.666231 0.007739 -1.994067 0.007739 -1.971109 2.666702 2.666664 1.434611e-05 -3.969306 1.434611e-05 -3.876288
N = 64 2.666666 2.671721 2.666638 0.001906 -1.998506 0.001906 -1.992582 2.666668 2.666666 8.567257e-07 -3.992241 8.567257e-07 -3.967753
N = 128 2.666666 2.667930 2.666664 0.000475 -1.999626 0.000475 -1.998132 2.666666 2.666666 5.289006e-08 -3.998054 5.289006e-08 -3.992024
N = 256 2.666666 2.666982 2.666666 0.000119 -1.999906 0.000119 -1.999532 2.666666 2.666666 2.955733e-09 -3.999619 2.955733e-09 -4.145738
N = 512 2.666666 2.666745 2.666666 0.000030 -1.999977 0.000030 -1.999883 2.666666 2.666666 1.072377e-09 -4.006796 1.072377e-09 -0.909712
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 0.25
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 2.666666 2.715719 NaN 1.839487e-02 NaN NaN NaN 2.714187 NaN 1.782020e-02 NaN NaN NaN
N = 8 2.666666 2.662150 NaN 1.693516e-03 -3.441210 NaN NaN 2.669460 NaN 1.047764e-03 -4.088127 NaN NaN
N = 16 2.666666 2.661848 2.661847 6.420173e-07 0.093302 6.420173e-07 NaN 2.666838 2.666675 6.121488e-05 -4.018859 6.121488e-05 NaN
N = 32 2.666666 2.664898 2.662123 1.042592e-03 -1.446493 1.042592e-03 3.337586 2.666677 2.666666 3.981784e-06 -4.004467 3.981784e-06 -4.019811
N = 64 2.666666 2.666147 2.667012 3.244563e-04 -1.767683 3.244563e-04 -1.288577 2.666667 2.666666 2.508414e-07 -4.001100 2.508414e-07 -4.004687
N = 128 2.666666 2.666526 2.666692 6.204964e-05 -1.891986 6.204964e-05 -1.718994 2.666666 2.666666 1.572281e-08 -4.000269 1.572281e-08 -3.999707
N = 256 2.666666 2.666630 2.666669 1.460619e-05 -1.947776 1.460619e-05 -1.871943 2.666666 2.666666 1.020082e-09 -4.000371 1.020082e-09 -3.948992
N = 512 2.666666 2.666657 2.666666 3.577514e-06 -1.974309 3.577514e-06 -1.938606 2.666666 2.666666 7.946340e-09 -4.026006 7.946340e-09 0.177873
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 0.5
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 8.438605 9.674633 NaN 0.146473 NaN NaN NaN 8.503685 NaN 7.712231e-03 NaN NaN NaN
N = 8 8.438605 8.767916 NaN 0.039024 -1.908190 NaN NaN 8.443410 NaN 5.694530e-04 -3.759500 NaN NaN
N = 16 8.438605 8.522370 8.431179 0.010816 -1.975035 0.010816 NaN 8.438919 8.438557 4.285047e-05 -3.935125 4.285047e-05 NaN
N = 32 8.438605 8.459639 8.438114 0.002551 -1.993614 0.002551 -1.968752 8.438625 8.438604 2.445375e-06 -3.983452 2.445375e-06 -3.931805
N = 64 8.438605 8.443869 8.438574 0.000628 -1.998394 0.000628 -1.992015 8.438606 8.438605 1.489725e-07 -3.995842 1.489725e-07 -3.982625
N = 128 8.438605 8.439921 8.438603 0.000156 -1.999598 0.000156 -1.997993 8.438605 8.438605 9.219334e-09 -3.998990 9.219334e-09 -4.000064
N = 256 8.438605 8.438934 8.438605 0.000039 -1.999899 0.000039 -1.999497 8.438605 8.438605 7.023049e-10 -4.001623 7.023049e-10 -3.735251
N = 512 8.438605 8.438687 8.438605 0.000010 -1.999975 0.000010 -1.999874 8.438605 8.438605 2.961137e-08 -4.057284 2.961137e-08 -0.026140
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 0.5
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 8.438605 8.652638 NaN 0.025364 NaN NaN NaN 8.463770 NaN 2.982193e-03 NaN NaN NaN
N = 8 8.438605 8.432466 NaN 0.000727 -5.123736 NaN NaN 8.440137 NaN 1.816114e-04 -4.037447 NaN NaN
N = 16 8.438605 8.429022 8.428967 0.000006 0.642537 0.000006 NaN 8.438700 8.438607 1.102988e-05 -4.008491 1.102988e-05 NaN
N = 32 8.438605 8.435145 8.431226 0.000465 -1.469764 0.000465 0.830074 8.438611 8.438605 7.006905e-07 -4.002060 7.006905e-07 -4.008915
N = 64 8.438605 8.437603 8.439252 0.000195 -1.788181 0.000195 -1.316727 8.438605 8.438605 4.395466e-08 -4.000510 4.395466e-08 -4.001907
N = 128 8.438605 8.438337 8.438650 0.000037 -1.903464 0.000037 -1.743720 8.438605 8.438605 2.756925e-09 -4.000105 2.756925e-09 -3.996608
N = 256 8.438605 8.438536 8.438609 0.000009 -1.953754 0.000009 -1.885546 8.438605 8.438605 9.335272e-09 -4.006893 9.335272e-09 -0.351898
N = 512 8.438605 8.438587 8.438605 0.000002 -1.977347 0.000002 -1.945635 8.438605 8.438605 3.033763e-08 -4.109874 3.033763e-08 0.007683
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 1.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 19.738549 20.447284 NaN 0.035906 NaN NaN NaN 19.752762 NaN 7.200901e-04 NaN NaN NaN
N = 8 19.738549 19.922720 NaN 0.009331 -1.944198 NaN NaN 19.739499 NaN 4.813553e-05 -3.903003 NaN NaN
N = 16 19.738549 19.785060 19.736081 0.002482 -1.985391 0.002482 NaN 19.738609 19.738545 3.241007e-06 -3.975074 3.241007e-06 NaN
N = 32 19.738549 19.750206 19.738390 0.000599 -1.996303 0.000599 -1.981722 19.738553 19.738549 1.949892e-07 -3.993725 1.949892e-07 -3.973832
N = 64 19.738549 19.741465 19.738539 0.000148 -1.999073 0.000148 -1.995378 19.738549 19.738549 1.196502e-08 -3.998427 1.196502e-08 -4.004838
N = 128 19.738549 19.739278 19.738548 0.000037 -1.999768 0.000037 -1.998841 19.738549 19.738549 7.604435e-10 -3.999475 7.604435e-10 -3.788746
N = 256 19.738549 19.738731 19.738549 0.000009 -1.999942 0.000009 -1.999710 19.738549 19.738549 1.678576e-08 -4.027237 1.678576e-08 0.061951
N = 512 19.738549 19.738594 19.738549 0.000002 -1.999986 0.000002 -1.999924 19.738549 19.738549 3.410828e-09 -3.453726 3.410828e-09 -0.017611
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 1.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 19.738549 19.679634 NaN 0.002985 NaN NaN NaN 19.746583 NaN 4.070439e-04 NaN NaN NaN
N = 8 19.738549 19.641717 NaN 0.004906 0.716842 NaN NaN 19.739039 NaN 2.483155e-05 -4.034938 NaN NaN
N = 16 19.738549 19.703350 19.665192 0.001940 -1.459966 0.001940 NaN 19.738579 19.738549 1.511130e-06 -4.008826 1.511130e-06 NaN
N = 32 19.738549 19.728328 19.745349 0.000862 -1.784051 0.000862 -1.303049 19.738551 19.738549 9.574890e-08 -4.002212 9.574890e-08 -4.009400
N = 64 19.738549 19.735813 19.739016 0.000162 -1.901696 0.000162 -1.738548 19.738549 19.738549 6.080926e-09 -4.000546 6.080926e-09 -3.985562
N = 128 19.738549 19.737842 19.738597 0.000038 -1.952962 0.000038 -1.883408 19.738549 19.738549 5.916877e-10 -4.000085 5.916877e-10 -3.396266
N = 256 19.738549 19.738369 19.738554 0.000009 -1.976979 0.000009 -1.944692 19.738549 19.738548 1.464429e-08 -4.036310 1.464429e-08 -0.034358
N = 512 19.738549 19.738504 19.738549 0.000002 -1.988614 0.000002 -1.973033 19.738549 19.738557 4.197772e-07 -4.681374 4.197772e-07 0.000076
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 2.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 36.441301 36.351342 NaN 2.468603e-03 NaN NaN NaN 36.420043 NaN 5.833573e-04 NaN NaN NaN
N = 8 36.441301 36.412116 NaN 8.008765e-04 -1.624043 NaN NaN 36.439800 NaN 4.120741e-05 -3.823404 NaN NaN
N = 16 36.441301 36.433533 36.445188 3.197816e-04 -1.909614 3.197816e-04 NaN 36.441204 36.441312 2.950879e-06 -3.954542 2.950879e-06 NaN
N = 32 36.441301 36.439329 36.441479 5.900323e-05 -1.977577 5.900323e-05 -1.885733 36.441295 36.441301 1.720071e-07 -3.988550 1.720071e-07 -3.952224
N = 64 36.441301 36.440806 36.441312 1.387073e-05 -1.994404 1.387073e-05 -1.971895 36.441301 36.441301 1.057241e-08 -3.997120 1.057241e-08 -3.986035
N = 128 36.441301 36.441177 36.441302 3.416958e-06 -1.998602 3.416958e-06 -1.993002 36.441301 36.441301 5.361641e-10 -3.998868 5.361641e-10 -4.115557
N = 256 36.441301 36.441270 36.441301 8.511403e-07 -1.999650 8.511403e-07 -1.998241 36.441301 36.441300 2.333626e-08 -3.967098 2.333626e-08 0.038464
N = 512 36.441301 36.441294 36.441301 2.127102e-07 -1.999937 2.127102e-07 -1.998956 36.441301 36.441308 1.884169e-07 -2.798008 1.884169e-07 -0.000275
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 2.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 36.441301 35.832852 NaN 0.016697 NaN NaN NaN 36.440338 NaN 2.644585e-05 NaN NaN NaN
N = 8 36.441301 36.204196 NaN 0.006507 -1.359606 NaN NaN 36.441232 NaN 1.912466e-06 -3.789535 NaN NaN
N = 16 36.441301 36.370470 36.505289 0.003693 -1.743070 0.003693 NaN 36.441297 36.441302 1.406682e-07 -3.949079 1.406682e-07 NaN
N = 32 36.441301 36.422094 36.445339 0.000638 -1.882741 0.000638 -1.687447 36.441301 36.441301 8.056553e-09 -3.987367 8.056553e-09 -3.944881
N = 64 36.441301 36.436309 36.441710 0.000148 -1.943854 0.000148 -1.860649 36.441301 36.441301 4.410936e-10 -3.996880 4.410936e-10 -3.962608
N = 128 36.441301 36.440029 36.441348 0.000036 -1.972518 0.000036 -1.933921 36.441301 36.441302 2.726522e-08 -3.988805 2.726522e-08 -0.024043
N = 256 36.441301 36.440980 36.441307 0.000009 -1.986403 0.000009 -1.967801 36.441301 36.441285 4.388124e-07 -3.749232 4.388124e-07 0.000094
N = 512 36.441301 36.441221 36.441302 0.000002 -1.993237 0.000002 -1.984105 36.441301 36.441142 4.360262e-06 0.671980 4.360262e-06 0.000002
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 4.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 55.383747 54.738450 NaN 1.165138e-02 NaN NaN NaN 55.350493 NaN 6.004303e-04 NaN NaN NaN
N = 8 55.383747 55.211305 NaN 3.113580e-03 -1.903854 NaN NaN 55.381453 NaN 4.141831e-05 -3.857656 NaN NaN
N = 16 55.383747 55.339880 55.387898 8.669335e-04 -1.974909 8.669335e-04 NaN 55.383600 55.383760 2.888108e-06 -3.963085 2.888108e-06 NaN
N = 32 55.383747 55.372732 55.384006 2.035705e-04 -1.993656 2.035705e-04 -1.968568 55.383737 55.383747 1.707474e-07 -3.990684 1.707474e-07 -3.961178
N = 64 55.383747 55.380990 55.383763 5.006916e-05 -1.998410 5.006916e-05 -1.992066 55.383746 55.383747 1.052346e-08 -3.997669 1.052346e-08 -3.989239
N = 128 55.383747 55.383057 55.383748 1.246579e-05 -1.999602 1.246579e-05 -1.998012 55.383747 55.383747 3.603694e-10 -3.998908 3.603694e-10 -4.817891
N = 256 55.383747 55.383574 55.383747 3.113229e-06 -1.999900 3.113229e-06 -1.999503 55.383747 55.383746 1.414303e-08 -4.010329 1.414303e-08 0.063983
N = 512 55.383747 55.383704 55.383747 7.781141e-07 -1.999979 7.781141e-07 -1.999865 55.383747 55.383739 1.297991e-07 -4.028543 1.297991e-07 0.000424
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 4.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 55.383747 54.410234 NaN 0.017578 NaN NaN NaN 55.379824 NaN 7.082611e-05 NaN NaN NaN
N = 8 55.383747 55.070107 NaN 0.005663 -1.634093 NaN NaN 55.383492 NaN 4.592963e-06 -3.946784 NaN NaN
N = 16 55.383747 55.295589 55.412632 0.002112 -1.830954 0.002112 NaN 55.383731 55.383747 2.989977e-07 -3.986123 2.989977e-07 NaN
N = 32 55.383747 55.360424 55.386591 0.000472 -1.918370 0.000472 -1.798167 55.383746 55.383747 1.831004e-08 -3.996495 1.831004e-08 -3.985276
N = 64 55.383747 55.377752 55.384071 0.000114 -1.959856 0.000114 -1.903734 55.383747 55.383747 1.826820e-09 -3.999134 1.826820e-09 -3.367154
N = 128 55.383747 55.382227 55.383786 0.000028 -1.980090 0.000028 -1.952921 55.383747 55.383747 3.753217e-09 -3.997425 3.753217e-09 -0.360222
N = 256 55.383747 55.383364 55.383751 0.000007 -1.990085 0.000007 -1.976712 55.383747 55.383739 1.296623e-07 -3.951150 1.296623e-07 0.000740
N = 512 55.383747 55.383651 55.383747 0.000002 -1.995056 0.000002 -1.988421 55.383747 55.383907 2.899492e-06 -1.097392 2.899492e-06 -0.000003
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 8.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 72.105546 71.363029 NaN 1.029764e-02 NaN NaN NaN 72.076518 NaN 4.025792e-04 NaN NaN NaN
N = 8 72.105546 71.911316 NaN 2.693691e-03 -1.934658 NaN NaN 72.103555 NaN 2.761201e-05 -3.865905 NaN NaN
N = 16 72.105546 72.056411 72.108626 7.241098e-04 -1.982937 7.241098e-04 NaN 72.105419 72.105557 1.913110e-06 -3.965081 1.913110e-06 NaN
N = 32 72.105546 72.093226 72.105743 1.735887e-04 -1.995685 1.735887e-04 -1.978646 72.105538 72.105547 1.134920e-07 -3.991177 1.134920e-07 -3.963367
N = 64 72.105546 72.102464 72.105559 4.292014e-05 -1.998918 4.292014e-05 -1.994605 72.105546 72.105546 6.975706e-09 -3.997776 6.975706e-09 -3.994622
N = 128 72.105546 72.104776 72.105547 1.070002e-05 -1.999729 1.070002e-05 -1.998648 72.105546 72.105547 1.299462e-09 -3.998574 1.299462e-09 -2.589904
N = 256 72.105546 72.105354 72.105547 2.673124e-06 -1.999933 2.673124e-06 -1.999662 72.105546 72.105545 1.857040e-08 -3.932799 1.857040e-08 0.031962
N = 512 72.105546 72.105498 72.105546 6.682154e-07 -1.999972 6.682154e-07 -1.999828 72.105546 72.105562 2.200364e-07 -1.539356 2.200364e-07 -0.000123
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 8.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 72.105546 71.167084 NaN 0.013015 NaN NaN NaN 72.101833 NaN 5.150457e-05 NaN NaN NaN
N = 8 72.105546 71.823222 NaN 0.003915 -1.732945 NaN NaN 72.105307 NaN 3.321129e-06 -3.954955 NaN NaN
N = 16 72.105546 72.028173 72.121272 0.001291 -1.867448 0.001291 NaN 72.105531 72.105547 2.148524e-07 -3.988195 2.148524e-07 NaN
N = 32 72.105546 72.085293 72.107363 0.000306 -1.933672 0.000306 -1.843217 72.105546 72.105546 1.314747e-08 -3.997014 1.314747e-08 -3.992830
N = 64 72.105546 72.100365 72.105768 0.000075 -1.966785 0.000075 -1.922112 72.105546 72.105546 9.292971e-10 -3.999105 9.292971e-10 -3.830052
N = 128 72.105546 72.104236 72.105574 0.000019 -1.983374 0.000019 -1.961125 72.105546 72.105548 2.624584e-08 -3.997261 2.624584e-08 -0.041629
N = 256 72.105546 72.105217 72.105550 0.000005 -1.991682 0.000005 -1.980572 72.105546 72.105550 5.247377e-08 -4.811127 5.247377e-08 -0.001360
N = 512 72.105546 72.105464 72.105547 0.000001 -1.995831 0.000001 -1.990264 72.105546 72.105640 1.292537e-06 1.420703 1.292537e-06 0.000003
Convergence of FDM Temperature at x_bar = 0.500 and k_ratio = 16.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 84.066993 83.499938 NaN 6.745276e-03 NaN NaN NaN 84.047391 NaN 2.331775e-04 NaN NaN NaN
N = 8 84.066993 83.920263 NaN 1.745393e-03 -1.950326 NaN NaN 84.065651 NaN 1.596694e-05 -3.868269 NaN NaN
N = 16 84.066993 84.029979 84.068733 4.609836e-04 -1.986999 4.609836e-04 NaN 84.066907 84.067000 1.104246e-06 -3.965621 1.104246e-06 NaN
N = 32 84.066993 84.057718 84.067105 1.116594e-04 -1.996710 1.116594e-04 -1.983738 84.066988 84.066993 6.557651e-08 -3.991309 6.557651e-08 -3.963928
N = 64 84.066993 84.064673 84.067000 2.768125e-05 -1.999175 2.768125e-05 -1.995887 84.066993 84.066993 3.988095e-09 -3.997789 3.988095e-09 -4.009398
N = 128 84.066993 84.066413 84.066994 6.905557e-06 -1.999794 6.905557e-06 -1.998968 84.066993 84.066993 8.955373e-10 -4.000157 8.955373e-10 -2.381555
N = 256 84.066993 84.066848 84.066993 1.725477e-06 -1.999949 1.725477e-06 -1.999734 84.066993 84.066996 3.310265e-08 -3.987292 3.310265e-08 -0.010228
N = 512 84.066993 84.066957 84.066993 4.313023e-07 -1.999980 4.313023e-07 -1.999948 84.066993 84.067022 3.427462e-07 -1.022110 3.427462e-07 -0.000034
Convergence of FEM Temperature at x_bar = 0.500 and k_ratio = 16.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 84.066993 83.389648 NaN 8.057211e-03 NaN NaN NaN 84.064444 NaN 3.031942e-05 NaN NaN NaN
N = 8 84.066993 83.869497 NaN 2.349269e-03 -1.778069 NaN NaN 84.066829 NaN 1.952881e-06 -3.956566 NaN NaN
N = 16 84.066993 84.013509 84.075262 7.345061e-04 -1.884628 7.345061e-04 NaN 84.066983 84.066993 1.261802e-07 -3.988602 1.261802e-07 NaN
N = 32 84.066993 84.053063 84.068040 1.781610e-04 -1.940876 1.781610e-04 -1.864285 84.066993 84.066993 7.651180e-09 -3.997117 7.651180e-09 -4.006481
N = 64 84.066993 84.063437 84.067126 4.387892e-05 -1.970029 4.387892e-05 -1.930747 84.066993 84.066993 1.028087e-09 -3.999169 1.028087e-09 -2.591014
N = 128 84.066993 84.066095 84.067010 1.088523e-05 -1.984906 1.088523e-05 -1.964966 84.066993 84.066990 3.353068e-08 -3.996938 3.353068e-08 0.019564
N = 256 84.066993 84.066767 84.066995 2.710530e-06 -1.992425 2.710530e-06 -1.982372 84.066993 84.066957 4.318022e-07 -4.806789 4.318022e-07 0.000097
N = 512 84.066993 84.066937 84.066993 6.762546e-07 -1.996214 6.762546e-07 -1.991181 84.066993 84.066953 4.734119e-07 1.292403 4.734119e-07 -0.000005
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 0.0625
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 0.392138 0.964920 NaN 1.460667 NaN NaN NaN 0.458151 NaN 1.683420e-01 NaN NaN NaN
N = 8 0.392138 0.534322 NaN 0.362587 -2.010228 NaN NaN 0.398641 NaN 1.658442e-02 -3.343494 NaN NaN
N = 16 0.392138 0.427450 0.392169 0.089965 -2.009513 0.089965 NaN 0.392612 0.391933 1.733889e-03 -3.775895 1.733889e-03 NaN
N = 32 0.392138 0.400947 0.392207 0.022284 -2.003074 0.022284 -2.011646 0.392169 0.392133 8.993455e-05 -3.938252 8.993455e-05 -3.763853
N = 64 0.392138 0.394339 0.392144 0.005597 -2.000819 0.005597 -2.003825 0.392140 0.392138 5.175510e-06 -3.984440 5.175510e-06 -3.935083
N = 128 0.392138 0.392688 0.392138 0.001402 -2.000209 0.001402 -2.001022 0.392138 0.392138 3.157303e-07 -4.000641 3.157303e-07 -3.983349
N = 256 0.392138 0.392275 0.392138 0.000351 -2.000055 0.000351 -2.000260 0.392138 0.392138 1.959448e-08 -4.075163 1.959448e-08 -3.996810
N = 512 0.392138 0.392172 0.392138 0.000088 -2.000026 0.000088 -2.000065 0.392138 0.392138 1.347800e-09 -6.903970 1.347800e-09 -3.868545
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 0.0625
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 0.392138 0.149088 NaN 0.619806 NaN NaN NaN 0.442756 NaN 1.290824e-01 NaN NaN NaN
N = 8 0.392138 0.367363 NaN 0.063178 -3.294328 NaN NaN 0.394802 NaN 6.793787e-03 -4.247932 NaN NaN
N = 16 0.392138 0.386079 0.387834 0.004526 -2.031788 0.004526 NaN 0.392299 0.392161 3.515340e-04 -4.048947 3.515340e-04 NaN
N = 32 0.392138 0.390525 0.391911 0.003535 -1.909892 0.003535 -2.073563 0.392148 0.392138 2.470996e-05 -4.011164 2.470996e-05 -4.051410
N = 64 0.392138 0.391716 0.392151 0.001110 -1.934512 0.001110 -1.901067 0.392138 0.392138 1.577555e-06 -4.003611 1.577555e-06 -4.011665
N = 128 0.392138 0.392029 0.392142 0.000286 -1.962964 0.000286 -1.924564 0.392138 0.392138 9.904100e-08 -4.015325 9.904100e-08 -4.002930
N = 256 0.392138 0.392110 0.392138 0.000071 -1.980497 0.000071 -1.956965 0.392138 0.392138 6.265574e-09 -4.257532 6.265574e-09 -3.985877
N = 512 0.392138 0.392131 0.392138 0.000018 -1.989961 0.000018 -1.977299 0.392138 0.392138 9.583398e-10 -2.911770 9.583398e-10 -2.340668
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 0.125
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 1.973771 2.884832 NaN 0.461584 NaN NaN NaN 2.054428 NaN 4.086461e-02 NaN NaN NaN
N = 8 1.973771 2.217011 NaN 0.123236 -1.905166 NaN NaN 1.980577 NaN 3.448250e-03 -3.566916 NaN NaN
N = 16 1.973771 2.035705 1.968139 0.034330 -1.973570 0.034330 NaN 1.974235 1.973640 3.018150e-04 -3.872719 3.018150e-04 NaN
N = 32 1.973771 1.989328 1.973387 0.008078 -1.993183 0.008078 -1.966930 1.973800 1.973768 1.622337e-05 -3.966691 1.622337e-05 -3.866069
N = 64 1.973771 1.977665 1.973746 0.001985 -1.998282 0.001985 -1.991477 1.973773 1.973771 9.652899e-07 -3.991576 9.652899e-07 -3.965005
N = 128 1.973771 1.974744 1.973769 0.000494 -1.999570 0.000494 -1.997852 1.973771 1.973771 5.954558e-08 -3.997927 5.954558e-08 -3.990989
N = 256 1.973771 1.974014 1.973771 0.000123 -1.999892 0.000123 -1.999462 1.973771 1.973771 3.607388e-09 -4.000377 3.607388e-09 -4.035270
N = 512 1.973771 1.973832 1.973771 0.000031 -1.999973 0.000031 -1.999865 1.973771 1.973771 5.886205e-10 -3.994490 5.886205e-10 -2.785349
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 0.125
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 1.973771 1.950437 NaN 0.011822 NaN NaN NaN 2.014040 NaN 2.040236e-02 NaN NaN NaN
N = 8 1.973771 1.953971 NaN 0.010032 -0.236949 NaN NaN 1.976128 NaN 1.194381e-03 -4.094401 NaN NaN
N = 16 1.973771 1.965537 1.948881 0.008546 -1.265896 0.008546 NaN 1.973916 1.973779 6.944648e-05 -4.019974 6.944648e-05 NaN
N = 32 1.973771 1.971213 1.976683 0.002767 -1.686822 0.002767 -1.026937 1.973780 1.973771 4.531587e-06 -4.004707 4.531587e-06 -4.020982
N = 64 1.973771 1.973063 1.973958 0.000453 -1.854433 0.000453 -1.617231 1.973771 1.973771 2.856214e-07 -4.001166 2.856214e-07 -4.004940
N = 128 1.973771 1.973585 1.973790 0.000104 -1.929744 0.000104 -1.826653 1.973771 1.973771 1.783865e-08 -4.000415 1.783865e-08 -4.004762
N = 256 1.973771 1.973723 1.973773 0.000025 -1.965480 0.000025 -1.917237 1.973771 1.973771 9.406020e-10 -4.002193 9.406020e-10 -4.235639
N = 512 1.973771 1.973759 1.973771 0.000006 -1.982889 0.000006 -1.959536 1.973771 1.973771 2.906984e-09 -4.013522 2.906984e-09 0.644742
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 0.25
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 6.560619 7.452341 NaN 0.135920 NaN NaN NaN 6.606093 NaN 6.931221e-03 NaN NaN NaN
N = 8 6.560619 6.799021 NaN 0.036338 -1.903201 NaN NaN 6.564006 NaN 5.161932e-04 -3.747126 NaN NaN
N = 16 6.560619 6.621332 6.554951 0.010127 -1.973315 0.010127 NaN 6.560841 6.560584 3.921835e-05 -3.931271 3.921835e-05 NaN
N = 32 6.560619 6.575870 6.560239 0.002383 -1.993146 0.002383 -1.966601 6.560633 6.560619 2.229184e-06 -3.982433 2.229184e-06 -3.927750
N = 64 6.560619 6.564437 6.560595 0.000586 -1.998275 0.000586 -1.991430 6.560620 6.560619 1.356016e-07 -3.995583 1.356016e-07 -3.981561
N = 128 6.560619 6.561574 6.560618 0.000146 -1.999568 0.000146 -1.997843 6.560619 6.560619 8.425880e-09 -3.998939 8.425880e-09 -3.993782
N = 256 6.560619 6.560858 6.560619 0.000036 -1.999892 0.000036 -1.999460 6.560619 6.560619 1.339557e-09 -4.001360 1.339557e-09 -2.781327
N = 512 6.560619 6.560679 6.560619 0.000009 -1.999973 0.000009 -1.999865 6.560619 6.560619 1.112903e-08 -3.955342 1.112903e-08 -0.062203
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 0.25
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 6.560619 6.572238 NaN 0.001771 NaN NaN NaN 6.581686 NaN 3.211104e-03 NaN NaN NaN
N = 8 6.560619 6.501798 NaN 0.008966 2.339890 NaN NaN 6.561891 NaN 1.938194e-04 -4.050285 NaN NaN
N = 16 6.560619 6.537168 6.525344 0.001812 -1.326678 0.001812 NaN 6.560698 6.560622 1.165695e-05 -4.011875 1.165695e-05 NaN
N = 32 6.560619 6.553589 6.567820 0.002167 -1.738076 0.002167 -1.106969 6.560624 6.560619 7.441118e-07 -4.002917 7.441118e-07 -4.012469
N = 64 6.560619 6.558712 6.561034 0.000354 -1.881813 0.000354 -1.680666 6.560620 6.560619 4.675026e-08 -4.000727 4.675026e-08 -4.002539
N = 128 6.560619 6.560124 6.560661 0.000082 -1.943682 0.000082 -1.859435 6.560619 6.560619 2.986564e-09 -4.000190 2.986564e-09 -3.972231
N = 256 6.560619 6.560493 6.560624 0.000020 -1.972494 0.000020 -1.933696 6.560619 6.560619 5.124541e-09 -4.004226 5.124541e-09 -0.618512
N = 512 6.560619 6.560588 6.560620 0.000005 -1.986406 0.000005 -1.967768 6.560619 6.560619 5.223584e-08 -3.907740 5.223584e-08 -0.004692
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 0.5
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 16.08921 16.377482 NaN 0.017917 NaN NaN NaN 16.072588 NaN 1.033089e-03 NaN NaN NaN
N = 8 16.08921 16.157318 NaN 0.004233 -2.081543 NaN NaN 16.087851 NaN 8.444136e-05 -3.612871 NaN NaN
N = 16 16.08921 16.105900 16.090233 0.000974 -2.028836 0.000974 NaN 16.089119 16.089234 7.136454e-06 -3.901042 7.136454e-06 NaN
N = 32 16.08921 16.093360 16.089315 0.000251 -2.007904 0.000251 -2.035696 16.089204 16.089210 3.811676e-07 -3.975106 3.811676e-07 -3.895879
N = 64 16.08921 16.090246 16.089218 0.000064 -2.002023 0.000064 -2.009856 16.089210 16.089210 2.290596e-08 -3.993769 2.290596e-08 -3.973248
N = 128 16.08921 16.089469 16.089211 0.000016 -2.000509 0.000016 -2.002527 16.089210 16.089210 2.318551e-09 -3.998132 2.318551e-09 -3.339367
N = 256 16.08921 16.089275 16.089210 0.000004 -2.000127 0.000004 -2.000635 16.089210 16.089210 2.682228e-09 -4.007949 2.682228e-09 0.981833
N = 512 16.08921 16.089226 16.089210 0.000001 -2.000027 0.000001 -2.000139 16.089210 16.089212 1.419220e-07 -3.851093 1.419220e-07 -0.000830
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 0.5
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 16.08921 15.648397 NaN 0.027398 NaN NaN NaN 16.093157 NaN 2.453157e-04 NaN NaN NaN
N = 8 16.08921 15.873740 NaN 0.013392 -1.032680 NaN NaN 16.089426 NaN 1.343464e-05 -4.190611 NaN NaN
N = 16 16.08921 16.020446 16.294141 0.016797 -1.647759 0.016797 NaN 16.089223 16.089211 7.271523e-07 -4.055423 7.271523e-07 NaN
N = 32 16.08921 16.070057 16.095406 0.001575 -1.844072 0.001575 -1.564197 16.089211 16.089210 4.840807e-08 -4.014442 4.840807e-08 -4.058229
N = 64 16.08921 16.084171 16.089784 0.000349 -1.926462 0.000349 -1.813482 16.089210 16.089210 2.737377e-09 -4.003653 2.737377e-09 -4.180015
N = 128 16.08921 16.087919 16.089273 0.000084 -1.964289 0.000084 -1.913195 16.089210 16.089210 3.562759e-09 -4.000903 3.562759e-09 2.475812
N = 256 16.08921 16.088883 16.089218 0.000021 -1.982405 0.000021 -1.958100 16.089210 16.089210 3.995362e-09 -3.978635 3.995362e-09 -0.064383
N = 512 16.08921 16.089128 16.089211 0.000005 -1.991265 0.000005 -1.979413 16.089210 16.089220 6.038491e-07 -3.906710 6.038491e-07 0.000031
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 1.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 31.182398 30.514477 NaN 0.021420 NaN NaN NaN 31.111549 NaN 2.272105e-03 NaN NaN NaN
N = 8 31.182398 30.989359 NaN 0.006191 -1.790787 NaN NaN 31.177173 NaN 1.675687e-04 -3.761205 NaN NaN
N = 16 31.182398 31.132234 31.193717 0.001971 -1.944144 0.001971 NaN 31.182057 31.182450 1.259443e-05 -3.937174 1.259443e-05 NaN
N = 32 31.182398 31.169733 31.183078 0.000428 -1.985785 0.000428 -1.929804 31.182377 31.182399 7.174384e-07 -3.984083 7.174384e-07 -3.933953
N = 64 31.182398 31.179224 31.182440 0.000103 -1.996430 0.000103 -1.982208 31.182397 31.182398 4.374637e-08 -3.996006 4.374637e-08 -3.982923
N = 128 31.182398 31.181604 31.182401 0.000026 -1.999106 0.000026 -1.995536 31.182398 31.182398 2.606778e-09 -3.998900 2.606778e-09 -4.051425
N = 256 31.182398 31.182200 31.182398 0.000006 -1.999777 0.000006 -1.998883 31.182398 31.182398 3.856539e-09 -3.998694 3.856539e-09 1.550952
N = 512 31.182398 31.182349 31.182398 0.000002 -1.999941 0.000002 -1.999717 31.182398 31.182398 1.478152e-08 -3.697036 1.478152e-08 0.015351
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 1.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 31.182398 29.932292 NaN 0.040090 NaN NaN NaN 31.175417 NaN 2.238893e-04 NaN NaN NaN
N = 8 31.182398 30.733838 NaN 0.014385 -1.478678 NaN NaN 31.181922 NaN 1.527575e-05 -3.873470 NaN NaN
N = 16 31.182398 31.050756 31.258002 0.006630 -1.768681 0.006630 NaN 31.182368 31.182401 1.052223e-06 -3.967146 1.052223e-06 NaN
N = 32 31.182398 31.146901 31.188771 0.001342 -1.890838 0.001342 -1.720830 31.182396 31.182398 6.260357e-08 -3.991707 6.260357e-08 -3.965417
N = 64 31.182398 31.173192 31.183087 0.000317 -1.947032 0.000317 -1.870631 31.182398 31.182398 3.987232e-09 -3.997921 3.987232e-09 -3.948103
N = 128 31.182398 31.180055 31.182479 0.000078 -1.973922 0.000078 -1.937733 31.182398 31.182398 3.060232e-09 -3.998459 3.060232e-09 2.496761
N = 256 31.182398 31.181807 31.182408 0.000019 -1.987063 0.000019 -1.969462 31.182398 31.182398 1.082074e-08 -4.017430 1.082074e-08 0.030395
N = 512 31.182398 31.182250 31.182400 0.000005 -1.993556 0.000005 -1.984879 31.182398 31.182407 2.634205e-07 -4.040315 2.634205e-07 -0.000086
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 2.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 49.706164 48.318359 NaN 0.027920 NaN NaN NaN 49.611795 NaN 1.898533e-03 NaN NaN NaN
N = 8 49.706164 49.325368 NaN 0.007661 -1.865716 NaN NaN 49.699312 NaN 1.378513e-04 -3.783700 NaN NaN
N = 16 49.706164 49.608556 49.719351 0.002228 -1.963950 0.002228 NaN 49.705718 49.706224 1.017998e-05 -3.942655 1.017998e-05 NaN
N = 32 49.706164 49.681606 49.707000 0.000511 -1.990813 0.000511 -1.954806 49.706136 49.706165 5.854912e-07 -3.985442 5.854912e-07 -3.939727
N = 64 49.706164 49.700015 49.706216 0.000125 -1.997692 0.000125 -1.988508 49.706162 49.706164 3.576368e-08 -3.996343 3.576368e-08 -3.984994
N = 128 49.706164 49.704626 49.706167 0.000031 -1.999422 0.000031 -1.997114 49.706164 49.706164 1.928803e-09 -3.999244 1.928803e-09 -4.188806
N = 256 49.706164 49.705779 49.706164 0.000008 -1.999856 0.000008 -1.999278 49.706164 49.706164 3.469626e-09 -3.992717 3.469626e-09 -0.677194
N = 512 49.706164 49.706068 49.706164 0.000002 -1.999965 0.000002 -1.999824 49.706164 49.706165 1.676751e-08 -4.252497 1.676751e-08 -0.011313
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 2.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 49.706164 47.861343 NaN 0.037115 NaN NaN NaN 49.694823 NaN 2.281547e-04 NaN NaN NaN
N = 8 49.706164 49.108745 NaN 0.012019 -1.626666 NaN NaN 49.705410 NaN 1.516824e-05 -3.910886 NaN NaN
N = 16 49.706164 49.536981 49.760850 0.004499 -1.820156 0.004499 NaN 49.706116 49.706166 1.015037e-06 -3.976421 1.015037e-06 NaN
N = 32 49.706164 49.661198 49.711952 0.001021 -1.911699 0.001021 -1.785533 49.706161 49.706164 6.131217e-08 -3.994020 6.131217e-08 -3.975493
N = 64 49.706164 49.694576 49.706841 0.000247 -1.956261 0.000247 -1.895902 49.706164 49.706164 3.568787e-09 -3.998501 3.568787e-09 -4.077754
N = 128 49.706164 49.703223 49.706246 0.000061 -1.978235 0.000061 -1.948711 49.706164 49.706164 1.470481e-09 -4.000422 1.470481e-09 -1.770777
N = 256 49.706164 49.705423 49.706174 0.000015 -1.989144 0.000015 -1.974544 49.706164 49.706166 4.850477e-08 -3.922125 4.850477e-08 -0.006553
N = 512 49.706164 49.705978 49.706165 0.000004 -1.994578 0.000004 -1.987318 49.706164 49.706153 2.272247e-07 -5.434926 2.272247e-07 0.000101
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 4.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 67.384857 65.895105 NaN 0.022108 NaN NaN NaN 67.299703 NaN 1.263688e-03 NaN NaN NaN
N = 8 67.384857 66.985431 NaN 0.005928 -1.899072 NaN NaN 67.378702 NaN 9.133429e-05 -3.790340 NaN NaN
N = 16 67.384857 67.283107 67.394899 0.001659 -1.972907 0.001659 NaN 67.384457 67.384909 6.709802e-06 -3.944130 6.709802e-06 NaN
N = 32 67.384857 67.359297 67.385506 0.000389 -1.993096 0.000389 -1.966071 67.384831 67.384858 3.870843e-07 -3.985796 3.870843e-07 -3.941284
N = 64 67.384857 67.378459 67.384898 0.000096 -1.998265 0.000096 -1.991365 67.384855 67.384857 2.370051e-08 -3.996432 2.370051e-08 -3.982999
N = 128 67.384857 67.383257 67.384859 0.000024 -1.999566 0.000024 -1.997832 67.384857 67.384857 1.671876e-09 -3.999119 1.671876e-09 -3.823551
N = 256 67.384857 67.384457 67.384857 0.000006 -1.999891 0.000006 -1.999458 67.384857 67.384856 3.788890e-09 -3.982202 3.788890e-09 0.650219
N = 512 67.384857 67.384757 67.384857 0.000001 -1.999966 0.000001 -1.999863 67.384857 67.384869 1.874138e-07 -4.196962 1.874138e-07 -0.000676
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 4.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 67.384857 65.562359 NaN 0.027046 NaN NaN NaN 67.374295 NaN 1.567351e-04 NaN NaN NaN
N = 8 67.384857 66.822621 NaN 0.008344 -1.696669 NaN NaN 67.384157 NaN 1.037839e-05 -3.916674 NaN NaN
N = 16 67.384857 67.228445 67.421196 0.002859 -1.845831 0.002859 NaN 67.384812 67.384859 6.914151e-07 -3.977871 6.914151e-07 NaN
N = 32 67.384857 67.343581 67.389183 0.000677 -1.921976 0.000677 -1.817524 67.384854 67.384857 4.190551e-08 -3.994383 4.190551e-08 -3.975621
N = 64 67.384857 67.374253 67.385391 0.000165 -1.960712 0.000165 -1.908339 67.384857 67.384857 2.046963e-09 -3.998556 2.046963e-09 -4.316630
N = 128 67.384857 67.382169 67.384923 0.000041 -1.980281 0.000041 -1.954008 67.384857 67.384857 2.172310e-09 -4.000123 2.172310e-09 3.109563
N = 256 67.384857 67.384180 67.384865 0.000010 -1.990121 0.000010 -1.976956 67.384857 67.384855 3.177752e-08 -4.072990 3.177752e-08 0.006917
N = 512 67.384857 67.384687 67.384858 0.000003 -1.995056 0.000003 -1.988465 67.384857 67.384981 1.850506e-06 -5.906415 1.850506e-06 -0.000007
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 8.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 80.863285 79.714831 NaN 1.420242e-02 NaN NaN NaN 80.803072 NaN 7.446247e-04 NaN NaN NaN
N = 8 80.863285 80.559732 NaN 3.753903e-03 -1.919673 NaN NaN 80.858940 NaN 5.373535e-05 -3.792570 NaN NaN
N = 16 80.863285 80.786251 80.869227 1.026052e-03 -1.978385 1.026052e-03 NaN 80.863003 80.863321 3.940703e-06 -3.944551 3.940703e-06 NaN
N = 32 80.863285 80.843953 80.863675 2.438962e-04 -1.994488 2.438962e-04 -1.972949 80.863267 80.863285 2.276011e-07 -3.985891 2.276011e-07 -3.941695
N = 64 80.863285 80.858447 80.863310 6.013105e-05 -1.998615 6.013105e-05 -1.993108 80.863284 80.863285 1.389183e-08 -3.996466 1.389183e-08 -3.987572
N = 128 80.863285 80.862075 80.863286 1.497913e-05 -1.999653 1.497913e-05 -1.998269 80.863285 80.863285 6.440270e-10 -3.998829 6.440270e-10 -4.254628
N = 256 80.863285 80.862982 80.863285 3.741411e-06 -1.999913 3.741411e-06 -1.999568 80.863285 80.863284 1.479848e-08 -4.006559 1.479848e-08 0.081162
N = 512 80.863285 80.863209 80.863285 9.351445e-07 -1.999979 9.351445e-07 -1.999890 80.863285 80.863293 9.851203e-08 -4.963517 9.851203e-08 -0.000812
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 8.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 80.863285 79.498914 NaN 0.016873 NaN NaN NaN 80.855768 NaN 9.295800e-05 NaN NaN NaN
N = 8 80.863285 80.453254 NaN 0.005071 -1.734431 NaN NaN 80.862788 NaN 6.149609e-06 -3.918013 NaN NaN
N = 16 80.863285 80.750296 80.884534 0.001660 -1.859558 0.001660 NaN 80.863253 80.863286 4.092696e-07 -3.978196 4.092696e-07 NaN
N = 32 80.863285 80.833578 80.866025 0.000401 -1.927308 0.000401 -1.834600 80.863283 80.863285 2.482608e-08 -3.994463 2.482608e-08 -3.975571
N = 64 80.863285 80.855665 80.863637 0.000099 -1.962954 0.000099 -1.914803 80.863285 80.863285 1.084340e-09 -3.998642 1.084340e-09 -4.470206
N = 128 80.863285 80.861355 80.863330 0.000024 -1.981291 0.000024 -1.956682 80.863285 80.863285 1.012989e-09 -3.995944 1.012989e-09 1.261409
N = 256 80.863285 80.862799 80.863290 0.000006 -1.990597 0.000006 -1.978148 80.863285 80.863295 1.229648e-07 -4.071341 1.229648e-07 -0.001059
N = 512 80.863285 80.863163 80.863286 0.000002 -1.995285 0.000002 -1.989011 80.863285 80.863291 7.828902e-08 -5.162794 7.828902e-08 -0.000108
Convergence of FDM Temperature at x_bar = 0.637 and k_ratio = 16.0
True T FDM 2 T FDM 2 Extrap T FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 T FDM 4 Extrap T FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 89.525116 88.797458 NaN 8.127968e-03 NaN NaN NaN 89.488625 NaN 4.076008e-04 NaN NaN NaN
N = 8 89.525116 89.334419 NaN 2.130095e-03 -1.931977 NaN NaN 89.522484 NaN 2.939692e-05 -3.793420 NaN NaN
N = 16 89.525116 89.476830 89.528233 5.741568e-04 -1.981624 5.741568e-04 NaN 89.524945 89.525138 2.154388e-06 -3.944680 2.154388e-06 NaN
N = 32 89.525116 89.513005 89.525323 1.375906e-04 -1.995309 1.375906e-04 -1.977014 89.525105 89.525116 1.244935e-07 -3.985917 1.244935e-07 -3.941793
N = 64 89.525116 89.522085 89.525129 3.399383e-05 -1.998821 3.399383e-05 -1.994135 89.525115 89.525116 7.612812e-09 -3.996463 7.612812e-09 -3.985113
N = 128 89.525116 89.524358 89.525116 8.472662e-06 -1.999705 8.472662e-06 -1.998526 89.525116 89.525116 8.748161e-11 -3.998068 8.748161e-11 -6.320010
N = 256 89.525116 89.524926 89.525116 2.116535e-06 -1.999925 2.116535e-06 -1.999635 89.525116 89.525118 2.897972e-08 -4.004300 2.897972e-08 -0.021888
N = 512 89.525116 89.525068 89.525116 5.290376e-07 -1.999984 5.290376e-07 -1.999904 89.525116 89.525119 3.980727e-08 -1.948540 3.980727e-08 -0.000790
Convergence of FEM Temperature at x_bar = 0.637 and k_ratio = 16.0
True T FEM 2 T FEM 2 Extrap T FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 T FEM 4 Extrap T FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 89.525116 88.671094 NaN 9.539464e-03 NaN NaN NaN 89.520554 NaN 5.095274e-05 NaN NaN NaN
N = 8 89.525116 89.272101 NaN 2.826187e-03 -1.755051 NaN NaN 89.524814 NaN 3.369930e-06 -3.918369 NaN NaN
N = 16 89.525116 89.455749 89.536559 9.025348e-04 -1.866912 9.025348e-04 NaN 89.525096 89.525117 2.242150e-07 -3.978276 2.242150e-07 NaN
N = 32 89.525116 89.506913 89.526672 2.207033e-04 -1.930093 2.207033e-04 -1.843749 89.525114 89.525116 1.353063e-08 -3.994481 1.353063e-08 -3.982852
N = 64 89.525116 89.520450 89.525321 5.440306e-05 -1.964098 5.440306e-05 -1.918185 89.525116 89.525116 1.848159e-09 -3.998667 1.848159e-09 -2.964279
N = 128 89.525116 89.523934 89.525142 1.348824e-05 -1.981797 1.348824e-05 -1.958049 89.525116 89.525117 1.247827e-08 -4.003756 1.247827e-08 -0.088223
N = 256 89.525116 89.524818 89.525119 3.356780e-06 -1.990833 3.356780e-06 -1.978749 89.525116 89.525130 1.627852e-07 -4.078042 1.627852e-07 -0.000436
N = 512 89.525116 89.525041 89.525116 8.372148e-07 -1.995395 8.372148e-07 -1.989289 89.525116 89.525379 2.942796e-06 -4.742373 2.942796e-06 -0.000001
Section 3 Temp: Convergence of FDM and FEM Temp with Varying k Ratio
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 0.0625
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -1.241829 -2.323339 NaN 0.870901 NaN NaN NaN -1.489915 NaN 1.997754e-01 NaN NaN NaN
N = 8 -1.241829 -1.583037 NaN 0.274763 -1.664323 NaN NaN -1.276591 NaN 2.799300e-02 -2.835241 NaN NaN
N = 16 -1.241829 -1.335328 -1.210764 0.102880 -1.867617 0.102880 NaN -1.244804 -1.239238 4.491405e-03 -3.546385 4.491405e-03 NaN
N = 32 -1.241829 -1.265851 -1.238768 0.021863 -1.960577 0.021863 -1.834029 -1.242032 -1.241768 2.131856e-04 -3.867273 2.131856e-04 -3.519703
N = 64 -1.241829 -1.247878 -1.241605 0.005052 -1.989623 0.005052 -1.950668 -1.241842 -1.241827 1.136302e-05 -3.965316 1.136302e-05 -3.860317
N = 128 -1.241829 -1.243344 -1.241814 0.001232 -1.997371 0.001232 -1.987025 -1.241829 -1.241829 6.746119e-07 -3.991275 6.746119e-07 -3.963554
N = 256 -1.241829 -1.242207 -1.241828 0.000306 -1.999340 0.000306 -1.996713 -1.241829 -1.241829 4.157855e-08 -3.998542 4.157855e-08 -3.991048
N = 512 -1.241829 -1.241923 -1.241828 0.000076 -1.999835 0.000076 -1.999175 -1.241829 -1.241829 2.836237e-09 -4.011395 2.836237e-09 -3.875262
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 0.0625
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -1.241829 -2.386292 NaN 0.921596 NaN NaN NaN -1.488766 NaN 1.988502e-01 NaN NaN NaN
N = 8 -1.241829 -1.638864 NaN 0.319718 -1.527334 NaN NaN -1.276827 NaN 2.818331e-02 -2.818769 NaN NaN
N = 16 -1.241829 -1.364788 -1.206097 0.131575 -1.691080 0.131575 NaN -1.244914 -1.239256 4.565149e-03 -3.503830 4.565149e-03 NaN
N = 32 -1.241829 -1.276744 -1.235074 0.033739 -1.816262 0.033739 -1.638264 -1.242045 -1.241762 2.281106e-04 -3.829927 2.281106e-04 -3.475886
N = 64 -1.241829 -1.251195 -1.240750 0.008418 -1.898303 0.008418 -1.784977 -1.241843 -1.241827 1.243065e-05 -3.942524 1.243065e-05 -3.821757
N = 128 -1.241829 -1.244259 -1.241674 0.002081 -1.946248 0.002081 -1.881117 -1.241829 -1.241829 7.417125e-07 -3.978996 7.417125e-07 -3.940018
N = 256 -1.241829 -1.242448 -1.241808 0.000515 -1.972331 0.000515 -1.937219 -1.241829 -1.241829 4.576965e-08 -3.992154 4.576965e-08 -3.977715
N = 512 -1.241829 -1.241985 -1.241826 0.000128 -1.985958 0.000128 -1.967700 -1.241829 -1.241829 3.130100e-09 -4.007118 3.130100e-09 -3.864662
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 0.125
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -1.756437 -2.635211 NaN 0.500316 NaN NaN NaN -1.898178 NaN 8.069805e-02 NaN NaN NaN
N = 8 -1.756437 -2.012353 NaN 0.145701 -1.779824 NaN NaN -1.771468 NaN 8.557588e-03 -3.237258 NaN NaN
N = 16 -1.756437 -1.823780 -1.741900 0.047006 -1.926064 0.047006 NaN -1.757555 -1.755839 9.773508e-04 -3.748951 9.773508e-04 NaN
N = 32 -1.756437 -1.773512 -1.755242 0.010409 -1.979630 0.010409 -1.907407 -1.756511 -1.756426 4.829010e-05 -3.931668 4.829010e-05 -3.735225
N = 64 -1.756437 -1.760722 -1.756356 0.002486 -1.994770 0.002486 -1.974522 -1.756442 -1.756437 2.747175e-06 -3.982525 2.747175e-06 -3.928168
N = 128 -1.756437 -1.757509 -1.756432 0.000613 -1.998684 0.000613 -1.993462 -1.756438 -1.756437 1.671405e-07 -3.995610 1.671405e-07 -3.981650
N = 256 -1.756437 -1.756705 -1.756437 0.000153 -1.999670 0.000153 -1.998354 -1.756437 -1.756437 1.041028e-08 -3.998942 1.041028e-08 -3.990638
N = 512 -1.756437 -1.756504 -1.756437 0.000038 -1.999918 0.000038 -1.999588 -1.756437 -1.756437 1.383606e-09 -4.000125 1.383606e-09 -3.001915
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 0.125
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -1.756437 -2.724280 NaN 0.551026 NaN NaN NaN -1.897893 NaN 8.053547e-02 NaN NaN NaN
N = 8 -1.756437 -2.072823 NaN 0.180129 -1.613087 NaN NaN -1.771784 NaN 8.737232e-03 -3.204376 NaN NaN
N = 16 -1.756437 -1.849907 -1.733951 0.066874 -1.759120 0.066874 NaN -1.757613 -1.755819 1.021680e-03 -3.706141 1.021680e-03 NaN
N = 32 -1.756437 -1.782145 -1.752552 0.016886 -1.862264 0.016886 -1.717974 -1.756516 -1.756424 5.242384e-05 -3.901773 5.242384e-05 -3.691040
N = 64 -1.756437 -1.763204 -1.755855 0.004186 -1.925724 0.004186 -1.838899 -1.756442 -1.756437 3.015804e-06 -3.965659 3.015804e-06 -3.897299
N = 128 -1.756437 -1.758175 -1.756357 0.001035 -1.961334 0.001035 -1.913214 -1.756438 -1.756437 1.838668e-07 -3.986774 1.838668e-07 -3.964188
N = 256 -1.756437 -1.756878 -1.756427 0.000257 -1.980259 0.000257 -1.954852 -1.756437 -1.756437 1.141355e-08 -3.994454 1.141355e-08 -3.986272
N = 512 -1.756437 -1.756548 -1.756436 0.000064 -1.990024 0.000064 -1.976959 -1.756437 -1.756437 7.108086e-10 -3.997585 7.108086e-10 -3.996485
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 0.25
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -2.486946 -3.171745 NaN 0.275357 NaN NaN NaN -2.556595 NaN 2.800566e-02 NaN NaN NaN
N = 8 -2.486946 -2.674530 NaN 0.075427 -1.868146 NaN NaN -2.492907 NaN 2.396673e-03 -3.546613 NaN NaN
N = 16 -2.486946 -2.535136 -2.480834 0.021889 -1.960724 0.021889 NaN -2.487355 -2.486824 2.132094e-04 -3.867344 2.132094e-04 NaN
N = 32 -2.486946 -2.499080 -2.486500 0.005059 -1.989662 0.005059 -1.950854 -2.486972 -2.486944 1.136517e-05 -3.965333 1.136517e-05 -3.860392
N = 64 -2.486946 -2.489985 -2.486917 0.001234 -1.997380 0.001234 -1.987073 -2.486948 -2.486946 6.747542e-07 -3.991233 6.747542e-07 -3.963582
N = 128 -2.486946 -2.487706 -2.486944 0.000306 -1.999343 0.000306 -1.996725 -2.486946 -2.486946 4.159483e-08 -3.997803 4.159483e-08 -3.990827
N = 256 -2.486946 -2.487136 -2.486946 0.000076 -1.999836 0.000076 -1.999178 -2.486946 -2.486946 2.466753e-09 -3.999493 2.466753e-09 -4.064070
N = 512 -2.486946 -2.486994 -2.486946 0.000019 -1.999959 0.000019 -1.999794 -2.486946 -2.486946 4.903515e-10 -3.995488 4.903515e-10 -1.980183
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 0.25
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -2.486946 -3.277606 NaN 0.317924 NaN NaN NaN -2.557137 NaN 2.822377e-02 NaN NaN NaN
N = 8 -2.486946 -2.732243 NaN 0.098634 -1.688525 NaN NaN -2.493130 NaN 2.486681e-03 -3.504617 NaN NaN
N = 16 -2.486946 -2.556640 -2.473243 0.033720 -1.815438 0.033720 NaN -2.487381 -2.486814 2.281691e-04 -3.830158 2.281691e-04 NaN
N = 32 -2.486946 -2.505646 -2.484779 0.008398 -1.897969 0.008398 -1.783949 -2.486974 -2.486944 1.243751e-05 -3.942603 1.243751e-05 -3.822000
N = 64 -2.486946 -2.491799 -2.486637 0.002076 -1.946097 0.002076 -1.880714 -2.486948 -2.486946 7.421638e-07 -3.978987 7.421638e-07 -3.940106
N = 128 -2.486946 -2.488183 -2.486905 0.000514 -1.972258 0.000514 -1.937039 -2.486946 -2.486946 4.576309e-08 -3.991494 4.576309e-08 -3.978786
N = 256 -2.486946 -2.487258 -2.486941 0.000128 -1.985922 0.000128 -1.967615 -2.486946 -2.486946 2.933993e-09 -3.996286 2.933993e-09 -3.952648
N = 512 -2.486946 -2.487025 -2.486946 0.000032 -1.992908 0.000032 -1.983571 -2.486946 -2.486946 1.338225e-09 -3.995277 1.338225e-09 0.015005
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 0.5
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -3.529489 -4.044925 NaN 0.146037 NaN NaN NaN -3.559595 NaN 8.529659e-03 NaN NaN NaN
N = 8 -3.529489 -3.665089 NaN 0.038419 -1.926443 NaN NaN -3.531728 NaN 6.341705e-04 -3.749546 NaN NaN
N = 16 -3.529489 -3.563869 -3.527097 0.010426 -1.979723 0.010426 NaN -3.529636 -3.529466 4.809035e-05 -3.931843 4.809035e-05 NaN
N = 32 -3.529489 -3.538115 -3.529327 0.002490 -1.994793 0.002490 -1.974639 -3.529499 -3.529489 2.736381e-06 -3.982571 2.736381e-06 -3.928352
N = 64 -3.529489 -3.531648 -3.529479 0.000614 -1.998689 0.000614 -1.993491 -3.529490 -3.529489 1.664953e-07 -3.995618 1.664953e-07 -3.981696
N = 128 -3.529489 -3.530029 -3.529489 0.000153 -1.999672 0.000153 -1.998362 -3.529489 -3.529489 1.035257e-08 -3.998886 1.035257e-08 -3.992959
N = 256 -3.529489 -3.529624 -3.529489 0.000038 -1.999918 0.000038 -1.999590 -3.529489 -3.529489 1.143815e-09 -3.999052 1.143815e-09 -3.239972
N = 512 -3.529489 -3.529523 -3.529489 0.000010 -1.999979 0.000010 -1.999898 -3.529489 -3.529489 7.826466e-10 -3.962070 7.826466e-10 2.125343
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 0.5
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -3.529489 -4.151337 NaN 0.176186 NaN NaN NaN -3.560400 NaN 8.757825e-03 NaN NaN NaN
N = 8 -3.529489 -3.714639 NaN 0.052458 -1.747869 NaN NaN -3.531856 NaN 6.705386e-04 -3.707180 NaN NaN
N = 16 -3.529489 -3.580543 -3.521119 0.016876 -1.858611 0.016876 NaN -3.529648 -3.529462 5.246721e-05 -3.902139 5.246721e-05 NaN
N = 32 -3.529489 -3.542941 -3.528289 0.004153 -1.924207 0.004153 -1.834401 -3.529499 -3.529489 3.019131e-06 -3.965805 3.019131e-06 -3.897682
N = 64 -3.529489 -3.532945 -3.529326 0.001026 -1.960639 0.001026 -1.911393 -3.529490 -3.529489 1.840754e-07 -3.986834 1.840754e-07 -3.964387
N = 128 -3.529489 -3.530365 -3.529468 0.000254 -1.979926 0.000254 -1.954030 -3.529489 -3.529489 1.149619e-08 -3.994426 1.149619e-08 -3.978192
N = 256 -3.529489 -3.529710 -3.529487 0.000063 -1.989861 0.000063 -1.976568 -3.529489 -3.529489 1.754708e-09 -3.996694 1.754708e-09 -2.821280
N = 512 -3.529489 -3.529545 -3.529489 0.000016 -1.994905 0.000016 -1.988168 -3.529489 -3.529489 7.304833e-09 -3.985645 7.304833e-09 0.138062
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 1.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -4.985127 -5.363661 NaN 0.075933 NaN NaN NaN -4.997127 NaN 2.407224e-03 NaN NaN NaN
N = 8 -4.985127 -5.082312 NaN 0.019495 -1.961618 NaN NaN -4.985949 NaN 1.649282e-04 -3.867460 NaN NaN
N = 16 -4.985127 -5.009594 -4.984248 0.005085 -1.989885 0.005085 NaN -4.985179 -4.985122 1.141325e-05 -3.965359 1.141325e-05 NaN
N = 32 -4.985127 -4.991254 -4.985069 0.001241 -1.997436 0.001241 -1.987353 -4.985130 -4.985126 6.776447e-07 -3.991240 6.776447e-07 -3.963608
N = 64 -4.985127 -4.986659 -4.985123 0.000308 -1.999357 0.000308 -1.996795 -4.985127 -4.985127 4.178327e-08 -3.997804 4.178327e-08 -3.990514
N = 128 -4.985127 -4.985510 -4.985126 0.000077 -1.999839 0.000077 -1.999196 -4.985127 -4.985127 2.828357e-09 -3.999474 2.828357e-09 -3.885074
N = 256 -4.985127 -4.985222 -4.985126 0.000019 -1.999960 0.000019 -1.999799 -4.985127 -4.985127 3.892385e-12 -3.994823 3.892385e-12 -9.291580
N = 512 -4.985127 -4.985150 -4.985127 0.000005 -1.999990 0.000005 -1.999950 -4.985127 -4.985126 1.878544e-08 -4.146631 1.878544e-08 -0.011762
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 1.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -4.985127 -5.459208 NaN 0.095099 NaN NaN NaN -4.997547 NaN 2.491440e-03 NaN NaN NaN
N = 8 -4.985127 -5.122610 NaN 0.027579 -1.785873 NaN NaN -4.985999 NaN 1.751167e-04 -3.830591 NaN NaN
N = 16 -4.985127 -5.022303 -4.979722 0.008551 -1.886793 0.008551 NaN -4.985183 -4.985121 1.245343e-05 -3.942758 1.245343e-05 NaN
N = 32 -4.985127 -4.994808 -4.984424 0.002083 -1.941161 0.002083 -1.867152 -4.985130 -4.985126 7.431957e-07 -3.979049 7.431957e-07 -3.940265
N = 64 -4.985127 -4.987598 -4.985035 0.000514 -1.969937 0.000514 -1.931164 -4.985127 -4.985127 4.586722e-08 -3.991520 4.586722e-08 -3.977691
N = 128 -4.985127 -4.985751 -4.985115 0.000128 -1.984797 0.000128 -1.964878 -4.985127 -4.985127 3.217920e-09 -3.996280 3.217920e-09 -3.830308
N = 256 -4.985127 -4.985283 -4.985125 0.000032 -1.992354 0.000032 -1.982251 -4.985127 -4.985126 2.619162e-09 -3.995181 2.619162e-09 -1.012964
N = 512 -4.985127 -4.985166 -4.985126 0.000008 -1.996167 0.000008 -1.991076 -4.985127 -4.985127 2.627928e-08 -3.954802 2.627928e-08 0.009191
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 2.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -6.832006 -7.120389 NaN 0.042211 NaN NaN NaN -6.837657 NaN 8.271733e-04 NaN NaN NaN
N = 8 -6.832006 -6.905215 NaN 0.010716 -1.977901 NaN NaN -6.832382 NaN 5.498761e-05 -3.911011 NaN NaN
N = 16 -6.832006 -6.850381 -6.831629 0.002745 -1.994255 0.002745 NaN -6.832030 -6.832005 3.679505e-06 -3.976922 3.679505e-06 NaN
N = 32 -6.832006 -6.836605 -6.831982 0.000677 -1.998549 0.000677 -1.992819 -6.832008 -6.832006 2.221497e-07 -3.994176 2.221497e-07 -3.975751
N = 64 -6.832006 -6.833156 -6.832005 0.000169 -1.999636 0.000169 -1.998186 -6.832006 -6.832006 1.379322e-08 -3.998527 1.379322e-08 -3.990379
N = 128 -6.832006 -6.832294 -6.832006 0.000042 -1.999909 0.000042 -1.999545 -6.832006 -6.832006 1.862709e-09 -3.999120 1.862709e-09 -2.981877
N = 256 -6.832006 -6.832078 -6.832006 0.000011 -1.999977 0.000011 -1.999886 -6.832006 -6.832006 1.324195e-08 -3.970921 1.324195e-08 0.090221
N = 512 -6.832006 -6.832024 -6.832006 0.000003 -1.999997 0.000003 -1.999971 -6.832006 -6.832005 2.031033e-07 -2.712521 2.031033e-07 -0.000329
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 2.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -6.832006 -7.202967 NaN 0.054297 NaN NaN NaN -6.836734 NaN 6.919495e-04 NaN NaN NaN
N = 8 -6.832006 -6.938675 NaN 0.015613 -1.798121 NaN NaN -6.832322 NaN 4.628007e-05 -3.902204 NaN NaN
N = 16 -6.832006 -6.860626 -6.827918 0.004790 -1.898047 0.004790 NaN -6.832026 -6.832005 3.115230e-06 -3.965787 3.115230e-06 NaN
N = 32 -6.832006 -6.839422 -6.831512 0.001158 -1.948400 0.001158 -1.880015 -6.832007 -6.832006 1.899456e-07 -3.986822 1.899456e-07 -3.964396
N = 64 -6.832006 -6.833894 -6.831944 0.000285 -1.974008 0.000285 -1.939550 -6.832006 -6.832006 1.170627e-08 -3.994421 1.170627e-08 -3.996168
N = 128 -6.832006 -6.832482 -6.831998 0.000071 -1.986952 0.000071 -1.969614 -6.832006 -6.832006 4.876349e-10 -3.996829 4.876349e-10 -4.556528
N = 256 -6.832006 -6.832126 -6.832005 0.000018 -1.993462 0.000018 -1.984762 -6.832006 -6.832006 4.071656e-09 -3.990079 4.071656e-09 0.267031
N = 512 -6.832006 -6.832036 -6.832006 0.000004 -1.996727 0.000004 -1.992369 -6.832006 -6.832006 5.252254e-08 -4.462020 5.252254e-08 0.001327
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 4.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -8.797049 -9.060288 NaN 0.029924 NaN NaN NaN -8.802357 NaN 6.033795e-04 NaN NaN NaN
N = 8 -8.797049 -8.864016 NaN 0.007612 -1.974855 NaN NaN -8.797407 NaN 4.074671e-05 -3.888310 NaN NaN
N = 16 -8.797049 -8.813868 -8.796658 0.001956 -1.993338 0.001956 NaN -8.797072 -8.797047 2.774659e-06 -3.970790 2.774659e-06 NaN
N = 32 -8.797049 -8.801259 -8.797023 0.000481 -1.998309 0.000481 -1.991674 -8.797050 -8.797049 1.660966e-07 -3.992612 1.660966e-07 -3.969303
N = 64 -8.797049 -8.798102 -8.797047 0.000120 -1.999576 0.000120 -1.997886 -8.797049 -8.797049 1.028146e-08 -3.998156 1.028146e-08 -3.989594
N = 128 -8.797049 -8.797312 -8.797049 0.000030 -1.999894 0.000030 -1.999469 -8.797049 -8.797049 4.932746e-10 -3.998186 4.932746e-10 -4.202856
N = 256 -8.797049 -8.797115 -8.797049 0.000007 -1.999973 0.000007 -1.999867 -8.797049 -8.797049 1.765533e-09 -4.027343 1.765533e-09 -0.422141
N = 512 -8.797049 -8.797065 -8.797049 0.000002 -1.999997 0.000002 -1.999954 -8.797049 -8.797046 2.985972e-07 -3.904864 2.985972e-07 -0.000177
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 4.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -8.797049 -9.133348 NaN 0.038229 NaN NaN NaN -8.799070 NaN 2.297472e-04 NaN NaN NaN
N = 8 -8.797049 -8.894031 NaN 0.011024 -1.793956 NaN NaN -8.797180 NaN 1.494372e-05 -3.942436 NaN NaN
N = 16 -8.797049 -8.823091 -8.793202 0.003399 -1.896890 0.003399 NaN -8.797057 -8.797049 9.755775e-07 -3.979737 9.755775e-07 NaN
N = 32 -8.797049 -8.803797 -8.796590 0.000819 -1.948194 0.000819 -1.878506 -8.797049 -8.797049 6.012137e-08 -3.992123 6.012137e-08 -3.978619
N = 64 -8.797049 -8.798767 -8.796992 0.000202 -1.974011 0.000202 -1.939272 -8.797049 -8.797049 3.963173e-09 -3.996647 3.963173e-09 -3.914403
N = 128 -8.797049 -8.797482 -8.797042 0.000050 -1.986981 0.000050 -1.969609 -8.797049 -8.797049 2.397599e-09 -3.996510 2.397599e-09 1.127149
N = 256 -8.797049 -8.797158 -8.797048 0.000012 -1.993485 0.000012 -1.984794 -8.797049 -8.797048 6.950475e-08 -3.965261 6.950475e-08 -0.004532
N = 512 -8.797049 -8.797076 -8.797049 0.000003 -1.996745 0.000003 -1.992394 -8.797049 -8.797060 1.303768e-06 -1.647698 1.303768e-06 0.000022
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 8.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -10.486214 -10.779857 NaN 0.028003 NaN NaN NaN -10.493612 NaN 7.054714e-04 NaN NaN NaN
N = 8 -10.486214 -10.561319 NaN 0.007162 -1.967076 NaN NaN -10.486719 NaN 4.814601e-05 -3.873099 NaN NaN
N = 16 -10.486214 -10.505104 -10.485636 0.001857 -1.991276 0.001857 NaN -10.486246 -10.486211 3.317279e-06 -3.966863 3.317279e-06 NaN
N = 32 -10.486214 -10.490944 -10.486176 0.000455 -1.997786 0.000455 -1.989096 -10.486216 -10.486214 1.973876e-07 -3.991621 1.973876e-07 -3.965170
N = 64 -10.486214 -10.487397 -10.486211 0.000113 -1.999444 0.000113 -1.997232 -10.486214 -10.486214 1.213030e-08 -3.997866 1.213030e-08 -3.996212
N = 128 -10.486214 -10.486510 -10.486214 0.000028 -1.999861 0.000028 -1.999305 -10.486214 -10.486214 2.131372e-09 -3.997144 2.131372e-09 -2.661836
N = 256 -10.486214 -10.486288 -10.486214 0.000007 -1.999966 0.000007 -1.999826 -10.486214 -10.486214 4.274711e-09 -3.846239 4.274711e-09 -0.220515
N = 512 -10.486214 -10.486232 -10.486214 0.000002 -1.999975 0.000002 -1.999969 -10.486214 -10.486214 4.898006e-09 -0.513431 4.898006e-09 -0.004645
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 8.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -10.486214 -10.847776 NaN 0.034480 NaN NaN NaN -10.487582 NaN 1.304358e-04 NaN NaN NaN
N = 8 -10.486214 -10.590389 NaN 0.009935 -1.795226 NaN NaN -10.486302 NaN 8.398688e-06 -3.957032 NaN NaN
N = 16 -10.486214 -10.514219 -10.482202 0.003054 -1.895250 0.003054 NaN -10.486219 -10.486214 5.423604e-07 -3.986778 5.423604e-07 NaN
N = 32 -10.486214 -10.493478 -10.485717 0.000740 -1.946777 0.000740 -1.876758 -10.486214 -10.486214 3.340698e-08 -3.995622 3.340698e-08 -3.987629
N = 64 -10.486214 -10.488064 -10.486151 0.000182 -1.973142 0.000182 -1.937655 -10.486214 -10.486214 2.688212e-09 -3.998136 2.688212e-09 -3.653156
N = 128 -10.486214 -10.486681 -10.486206 0.000045 -1.986505 0.000045 -1.968604 -10.486214 -10.486214 6.990587e-09 -3.995967 6.990587e-09 -0.354642
N = 256 -10.486214 -10.486331 -10.486213 0.000011 -1.993235 0.000011 -1.984241 -10.486214 -10.486213 8.708136e-08 -5.849693 8.708136e-08 -0.002119
N = 512 -10.486214 -10.486243 -10.486214 0.000003 -1.996597 0.000003 -1.992109 -10.486214 -10.486224 9.472036e-07 3.111172 9.472036e-07 -0.000026
Convergence of FDM Heat with x_bar = 0.500 and k_ratio = 16.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -11.680625 -12.024326 NaN 0.029425 NaN NaN NaN -11.690359 NaN 8.333062e-04 NaN NaN NaN
N = 8 -11.680625 -11.768665 NaN 0.007537 -1.964919 NaN NaN -11.681291 NaN 5.699702e-05 -3.869888 NaN NaN
N = 16 -11.680625 -11.702777 -11.679901 0.001959 -1.990732 0.001959 NaN -11.680668 -11.680622 3.936877e-06 -3.966020 3.936877e-06 NaN
N = 32 -11.680625 -11.686172 -11.680577 0.000479 -1.997649 0.000479 -1.988414 -11.680628 -11.680625 2.339626e-07 -3.991409 2.339626e-07 -3.964284
N = 64 -11.680625 -11.682012 -11.680622 0.000119 -1.999410 0.000119 -1.997062 -11.680625 -11.680625 1.444745e-08 -3.997772 1.444745e-08 -3.989014
N = 128 -11.680625 -11.680972 -11.680625 0.000030 -1.999852 0.000030 -1.999263 -11.680625 -11.680625 1.394570e-09 -4.001136 1.394570e-09 -3.412796
N = 256 -11.680625 -11.680712 -11.680625 0.000007 -1.999964 0.000007 -1.999815 -11.680625 -11.680625 1.645965e-08 -3.981056 1.645965e-08 -0.071781
N = 512 -11.680625 -11.680647 -11.680625 0.000002 -1.999977 0.000002 -1.999958 -11.680625 -11.680624 1.365727e-07 0.019875 1.365727e-07 0.000008
Convergence of FEM Heat with x_bar = 0.500 and k_ratio = 16.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -11.680625 -12.090051 NaN 0.035052 NaN NaN NaN -11.682009 NaN 1.185027e-04 NaN NaN NaN
N = 8 -11.680625 -11.797909 NaN 0.010041 -1.803594 NaN NaN -11.680714 NaN 7.625382e-06 -3.957966 NaN NaN
N = 16 -11.680625 -11.712125 -11.676464 0.003054 -1.896582 0.003054 NaN -11.680631 -11.680625 4.921369e-07 -3.988478 4.921369e-07 NaN
N = 32 -11.680625 -11.688797 -11.680083 0.000746 -1.946658 0.000746 -1.878621 -11.680625 -11.680625 3.029433e-08 -3.996830 3.029433e-08 -3.987749
N = 64 -11.680625 -11.682707 -11.680555 0.000184 -1.972873 0.000184 -1.937587 -11.680625 -11.680625 1.396672e-09 -3.998848 1.396672e-09 -4.405824
N = 128 -11.680625 -11.681150 -11.680616 0.000046 -1.986316 0.000046 -1.968306 -11.680625 -11.680625 4.998899e-09 -3.994560 4.998899e-09 -0.436061
N = 256 -11.680625 -11.680757 -11.680624 0.000011 -1.993128 0.000011 -1.984024 -11.680625 -11.680624 9.229468e-08 -6.569744 9.229468e-08 -0.001825
N = 512 -11.680625 -11.680658 -11.680625 0.000003 -1.996572 0.000003 -1.991978 -11.680625 -11.680638 1.106963e-06 4.036521 1.106963e-06 -0.000025
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 0.0625
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -1.241978 -1.892412 NaN 0.523708 NaN NaN NaN -1.350272 NaN 8.719507e-02 NaN NaN NaN
N = 8 -1.241978 -1.432463 NaN 0.153372 -1.771728 NaN NaN -1.253695 NaN 9.433953e-03 -3.208312 NaN NaN
N = 16 -1.241978 -1.292229 -1.230720 0.049978 -1.922447 0.049978 NaN -1.242857 -1.241487 1.103433e-03 -3.736406 1.103433e-03 NaN
N = 32 -1.241978 -1.254729 -1.241041 0.011030 -1.978528 0.011030 -1.902870 -1.242036 -1.241968 5.422868e-05 -3.927976 5.422868e-05 -3.721927
N = 64 -1.241978 -1.245178 -1.241914 0.002628 -1.994480 0.002628 -1.973144 -1.241982 -1.241978 3.071666e-06 -3.982155 3.071666e-06 -3.924241
N = 128 -1.241978 -1.242779 -1.241974 0.000648 -1.998612 0.000648 -1.993098 -1.241978 -1.241978 1.866167e-07 -4.005072 1.866167e-07 -3.980600
N = 256 -1.241978 -1.242178 -1.241978 0.000161 -1.999661 0.000161 -1.998262 -1.241978 -1.241978 1.167171e-08 -4.163740 1.167171e-08 -3.984220
N = 512 -1.241978 -1.242028 -1.241978 0.000040 -1.999948 0.000040 -1.999565 -1.241978 -1.241978 1.129908e-09 -4.093738 1.129908e-09 -3.403237
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 0.0625
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -1.241978 -1.955601 NaN 0.574586 NaN NaN NaN -1.350006 NaN 8.698070e-02 NaN NaN NaN
N = 8 -1.241978 -1.476318 NaN 0.188683 -1.606560 NaN NaN -1.253926 NaN 9.620105e-03 -3.176571 NaN NaN
N = 16 -1.241978 -1.311448 -1.224995 0.070574 -1.754142 0.070574 NaN -1.242902 -1.241473 1.150974e-03 -3.693351 1.150974e-03 NaN
N = 32 -1.241978 -1.261128 -1.239024 0.017840 -1.859012 0.017840 -1.712139 -1.242040 -1.241967 5.881873e-05 -3.897482 5.881873e-05 -3.677491
N = 64 -1.241978 -1.247025 -1.241533 0.004424 -1.923832 0.004424 -1.835089 -1.241982 -1.241978 3.371315e-06 -3.964820 3.371315e-06 -3.892758
N = 128 -1.241978 -1.243275 -1.241917 0.001094 -1.960308 0.001094 -1.910998 -1.241978 -1.241978 2.052741e-07 -3.995169 2.052741e-07 -3.962755
N = 256 -1.241978 -1.242307 -1.241970 0.000271 -1.979729 0.000271 -1.953652 -1.241978 -1.241978 1.268251e-08 -4.143718 1.268251e-08 -3.991678
N = 512 -1.241978 -1.242061 -1.241977 0.000067 -1.989772 0.000067 -1.976334 -1.241978 -1.241978 1.753379e-09 -4.424299 1.753379e-09 -2.957447
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 0.125
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -1.758799 -2.268096 NaN 0.289571 NaN NaN NaN -1.812674 NaN 3.063204e-02 NaN NaN NaN
N = 8 -1.758799 -1.898803 NaN 0.079602 -1.863040 NaN NaN -1.763475 NaN 2.658650e-03 -3.526276 NaN NaN
N = 16 -1.758799 -1.794813 -1.754052 0.023238 -1.958837 0.023238 NaN -1.759121 -1.758698 2.403705e-04 -3.860338 2.403705e-04 NaN
N = 32 -1.758799 -1.767870 -1.758449 0.005358 -1.989132 0.005358 -1.948491 -1.758819 -1.758797 1.273760e-05 -3.963414 1.273760e-05 -3.853001
N = 64 -1.758799 -1.761071 -1.758776 0.001305 -1.997244 0.001305 -1.986411 -1.758800 -1.758799 7.542667e-07 -3.990746 7.542667e-07 -3.961560
N = 128 -1.758799 -1.759367 -1.758797 0.000324 -1.999308 0.000324 -1.996554 -1.758799 -1.758799 4.646868e-08 -3.997740 4.646868e-08 -3.990090
N = 256 -1.758799 -1.758941 -1.758799 0.000081 -1.999827 0.000081 -1.999136 -1.758799 -1.758799 2.749222e-09 -4.000295 2.749222e-09 -4.066683
N = 512 -1.758799 -1.758834 -1.758799 0.000020 -1.999957 0.000020 -1.999784 -1.758799 -1.758799 1.148633e-09 -4.020561 1.148633e-09 -0.442350
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 0.125
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -1.758799 -2.343653 NaN 0.332530 NaN NaN NaN -1.813018 NaN 3.082735e-02 NaN NaN NaN
N = 8 -1.758799 -1.940923 NaN 0.103550 -1.683153 NaN NaN -1.763642 NaN 2.753720e-03 -3.484757 NaN NaN
N = 16 -1.758799 -1.810685 -1.748438 0.035602 -1.811494 0.035602 NaN -1.759141 -1.758690 2.566924e-04 -3.822634 2.566924e-04 NaN
N = 32 -1.758799 -1.772745 -1.757148 0.008876 -1.895524 0.008876 -1.779334 -1.758821 -1.758797 1.392563e-05 -3.940173 1.392563e-05 -3.814075
N = 64 -1.758799 -1.762421 -1.758563 0.002194 -1.944724 0.002194 -1.877853 -1.758800 -1.758799 8.290856e-07 -3.978186 8.290856e-07 -3.937561
N = 128 -1.758799 -1.759723 -1.758767 0.000543 -1.971529 0.000543 -1.935435 -1.758799 -1.758799 5.110725e-08 -3.991256 5.110725e-08 -3.977434
N = 256 -1.758799 -1.759032 -1.758795 0.000135 -1.985546 0.000135 -1.966763 -1.758799 -1.758799 3.091361e-09 -3.997014 3.091361e-09 -4.030266
N = 512 -1.758799 -1.758857 -1.758798 0.000034 -1.992717 0.000034 -1.983132 -1.758799 -1.758799 3.903887e-11 -4.015170 3.903887e-11 -6.231455
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 0.25
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -2.501202 -2.888923 NaN 0.155014 NaN NaN NaN -2.524883 NaN 9.467815e-03 NaN NaN NaN
N = 8 -2.501202 -2.603336 NaN 0.040834 -1.924555 NaN NaN -2.502977 NaN 7.099598e-04 -3.737222 NaN NaN
N = 16 -2.501202 -2.527108 -2.499354 0.011105 -1.979071 0.011105 NaN -2.501318 -2.501182 5.435757e-05 -3.928158 5.435757e-05 NaN
N = 32 -2.501202 -2.507703 -2.501075 0.002650 -1.994616 0.002650 -1.973826 -2.501209 -2.501201 3.079948e-06 -3.981605 3.079948e-06 -3.924474
N = 64 -2.501202 -2.502828 -2.501194 0.000654 -1.998644 0.000654 -1.993269 -2.501202 -2.501202 1.871398e-07 -3.995374 1.871398e-07 -3.980634
N = 128 -2.501202 -2.501608 -2.501201 0.000163 -1.999660 0.000163 -1.998305 -2.501202 -2.501202 1.160713e-08 -3.998840 1.160713e-08 -3.995524
N = 256 -2.501202 -2.501303 -2.501202 0.000041 -1.999915 0.000041 -1.999576 -2.501202 -2.501202 3.385139e-10 -3.999523 3.385139e-10 -4.957265
N = 512 -2.501202 -2.501227 -2.501202 0.000010 -1.999979 0.000010 -1.999894 -2.501202 -2.501202 2.176780e-08 -4.006309 2.176780e-08 -0.044305
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 0.25
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -2.501202 -2.964641 NaN 0.185287 NaN NaN NaN -2.525373 NaN 9.664044e-03 NaN NaN NaN
N = 8 -2.501202 -2.639437 NaN 0.055267 -1.745259 NaN NaN -2.503068 NaN 7.461252e-04 -3.695137 NaN NaN
N = 16 -2.501202 -2.539384 -2.494922 0.017821 -1.856151 0.017821 NaN -2.501327 -2.501179 5.893199e-05 -3.898067 5.893199e-05 NaN
N = 32 -2.501202 -2.511273 -2.500289 0.004393 -1.922655 0.004393 -1.831559 -2.501210 -2.501201 3.379471e-06 -3.964507 3.379471e-06 -3.893406
N = 64 -2.501202 -2.503791 -2.501077 0.001085 -1.959773 0.001085 -1.909585 -2.501202 -2.501202 2.057958e-07 -3.986394 2.057958e-07 -3.963003
N = 128 -2.501202 -2.501858 -2.501185 0.000269 -1.979469 0.000269 -1.953020 -2.501202 -2.501202 1.273715e-08 -3.994267 1.273715e-08 -3.989443
N = 256 -2.501202 -2.501367 -2.501200 0.000067 -1.989627 0.000067 -1.976035 -2.501202 -2.501202 1.639429e-09 -3.997049 1.639429e-09 -3.045573
N = 512 -2.501202 -2.501243 -2.501201 0.000017 -1.994786 0.000017 -1.987895 -2.501202 -2.501202 9.024079e-09 -4.008757 9.024079e-09 -0.114514
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 0.5
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -3.564213 -3.858649 NaN 0.082609 NaN NaN NaN -3.574224 NaN 2.808695e-03 NaN NaN NaN
N = 8 -3.564213 -3.639854 NaN 0.021222 -1.960716 NaN NaN -3.564905 NaN 1.940781e-04 -3.855191 NaN NaN
N = 16 -3.564213 -3.583260 -3.563514 0.005541 -1.989586 0.005541 NaN -3.564257 -3.564209 1.355828e-05 -3.961911 1.355828e-05 NaN
N = 32 -3.564213 -3.568983 -3.564167 0.001351 -1.997356 0.001351 -1.986980 -3.564216 -3.564213 8.014377e-07 -3.990351 8.014377e-07 -3.959982
N = 64 -3.564213 -3.565406 -3.564210 0.000336 -1.999336 0.000336 -1.996695 -3.564213 -3.564213 4.932485e-08 -3.997581 4.932485e-08 -3.990263
N = 128 -3.564213 -3.564511 -3.564213 0.000084 -1.999834 0.000084 -1.999170 -3.564213 -3.564213 2.989209e-09 -3.999323 2.989209e-09 -4.034364
N = 256 -3.564213 -3.564287 -3.564213 0.000021 -1.999958 0.000021 -1.999792 -3.564213 -3.564213 4.437690e-09 -4.004355 4.437690e-09 -0.720881
N = 512 -3.564213 -3.564231 -3.564213 0.000005 -1.999990 0.000005 -1.999949 -3.564213 -3.564213 1.230940e-08 -3.892410 1.230940e-08 0.021047
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 0.5
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -3.564213 -3.926446 NaN 0.101631 NaN NaN NaN -3.574143 NaN 2.786130e-03 NaN NaN NaN
N = 8 -3.564213 -3.669207 NaN 0.029458 -1.786611 NaN NaN -3.564914 NaN 1.966313e-04 -3.824698 NaN NaN
N = 16 -3.564213 -3.592628 -3.560167 0.009118 -1.885584 0.009118 NaN -3.564258 -3.564208 1.404724e-05 -3.940985 1.404724e-05 NaN
N = 32 -3.564213 -3.571617 -3.563673 0.002229 -1.940239 0.002229 -1.865819 -3.564216 -3.564213 8.366621e-07 -3.978534 8.366621e-07 -3.938406
N = 64 -3.564213 -3.566104 -3.564142 0.000550 -1.969405 0.000550 -1.930100 -3.564213 -3.564213 5.159298e-08 -3.991362 5.159298e-08 -3.977450
N = 128 -3.564213 -3.564691 -3.564204 0.000137 -1.984514 0.000137 -1.964260 -3.564213 -3.564213 3.111179e-09 -3.996222 3.111179e-09 -4.034282
N = 256 -3.564213 -3.564333 -3.564212 0.000034 -1.992209 0.000034 -1.981922 -3.564213 -3.564213 5.160476e-09 -3.999598 5.160476e-09 -0.661124
N = 512 -3.564213 -3.564243 -3.564213 0.000008 -1.996091 0.000008 -1.990908 -3.564213 -3.564213 4.864156e-08 -3.819867 4.864156e-08 0.005524
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 1.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -4.985127 -5.229953 NaN 0.049111 NaN NaN NaN -4.991324 NaN 1.243210e-03 NaN NaN NaN
N = 8 -4.985127 -5.047653 NaN 0.012543 -1.969225 NaN NaN -4.985553 NaN 8.562086e-05 -3.859964 NaN NaN
N = 16 -4.985127 -5.000849 -4.984681 0.003243 -1.991676 0.003243 NaN -4.985154 -4.985124 5.959042e-06 -3.962550 5.959042e-06 NaN
N = 32 -4.985127 -4.989063 -4.985096 0.000796 -1.997874 0.000796 -1.989601 -4.985128 -4.985126 3.532359e-07 -3.990470 3.532359e-07 -3.960660
N = 64 -4.985127 -4.986111 -4.985125 0.000198 -1.999466 0.000198 -1.997343 -4.985127 -4.985127 2.177549e-08 -3.997603 2.177549e-08 -3.988623
N = 128 -4.985127 -4.985373 -4.985126 0.000049 -1.999866 0.000049 -1.999332 -4.985127 -4.985127 1.004778e-09 -3.999222 1.004778e-09 -4.404782
N = 256 -4.985127 -4.985188 -4.985127 0.000012 -1.999967 0.000012 -1.999833 -4.985127 -4.985127 1.382190e-08 -4.001567 1.382190e-08 0.138848
N = 512 -4.985127 -4.985142 -4.985127 0.000003 -1.999989 0.000003 -1.999960 -4.985127 -4.985127 9.249491e-09 -3.239600 9.249491e-09 0.011826
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 1.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -4.985127 -5.290994 NaN 0.061356 NaN NaN NaN -4.989179 NaN 8.128531e-04 NaN NaN NaN
N = 8 -4.985127 -5.073389 NaN 0.017705 -1.793026 NaN NaN -4.985398 NaN 5.448497e-05 -3.899064 NaN NaN
N = 16 -4.985127 -5.008894 -4.981727 0.005453 -1.892797 0.005453 NaN -4.985144 -4.985126 3.676566e-06 -3.965409 3.676566e-06 NaN
N = 32 -4.985127 -4.991298 -4.984696 0.001324 -1.945278 0.001324 -1.873929 -4.985128 -4.985126 2.237553e-07 -3.986976 2.237553e-07 -3.963947
N = 64 -4.985127 -4.986699 -4.985072 0.000326 -1.972350 0.000326 -1.935902 -4.985127 -4.985127 1.393229e-08 -3.994584 1.393229e-08 -3.981692
N = 128 -4.985127 -4.985524 -4.985120 0.000081 -1.986102 0.000081 -1.967677 -4.985127 -4.985127 8.685149e-11 -3.997184 8.685149e-11 -7.228264
N = 256 -4.985127 -4.985226 -4.985126 0.000020 -1.993033 0.000020 -1.983770 -4.985127 -4.985126 1.520828e-08 -4.005209 1.520828e-08 -0.074896
N = 512 -4.985127 -4.985152 -4.985126 0.000005 -1.996511 0.000005 -1.991868 -4.985127 -4.985126 1.693590e-07 -5.484669 1.693590e-07 -0.000448
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 2.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -6.610807 -6.872679 NaN 0.039613 NaN NaN NaN -6.619428 NaN 1.304078e-03 NaN NaN NaN
N = 8 -6.610807 -6.678556 NaN 0.010248 -1.950595 NaN NaN -6.611422 NaN 9.300092e-05 -3.809641 NaN NaN
N = 16 -6.610807 -6.627905 -6.610024 0.002705 -1.986349 0.002705 NaN -6.610847 -6.610802 6.730247e-06 -3.949086 6.730247e-06 NaN
N = 32 -6.610807 -6.615092 -6.610753 0.000656 -1.996494 0.000656 -1.982940 -6.610809 -6.610807 3.913050e-07 -3.987045 3.913050e-07 -3.946493
N = 64 -6.610807 -6.611879 -6.610804 0.000163 -1.999117 0.000163 -1.995618 -6.610807 -6.610807 2.399182e-08 -3.996738 2.399182e-08 -3.985134
N = 128 -6.610807 -6.611075 -6.610807 0.000041 -1.999779 0.000041 -1.998897 -6.610807 -6.610807 1.566345e-09 -3.999742 1.566345e-09 -3.929495
N = 256 -6.610807 -6.610874 -6.610807 0.000010 -1.999945 0.000010 -1.999724 -6.610807 -6.610807 8.518639e-09 -3.981280 8.518639e-09 -0.218471
N = 512 -6.610807 -6.610824 -6.610807 0.000003 -1.999987 0.000003 -1.999931 -6.610807 -6.610807 7.573610e-08 -4.739885 7.573610e-08 0.001727
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 2.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -6.610807 -6.934589 NaN 0.048978 NaN NaN NaN -6.613014 NaN 3.338215e-04 NaN NaN NaN
N = 8 -6.610807 -6.705769 NaN 0.014365 -1.769597 NaN NaN -6.610952 NaN 2.191865e-05 -3.928846 NaN NaN
N = 16 -6.610807 -6.636589 -6.606609 0.004538 -1.881011 0.004538 NaN -6.610816 -6.610807 1.446114e-06 -3.977416 1.446114e-06 NaN
N = 32 -6.610807 -6.617529 -6.610280 0.001097 -1.939466 0.001097 -1.859818 -6.610808 -6.610807 8.847010e-08 -3.992259 8.847010e-08 -3.976406
N = 64 -6.610807 -6.612523 -6.610741 0.000270 -1.969471 0.000270 -1.929031 -6.610807 -6.610807 5.449351e-09 -3.997041 5.449351e-09 -4.003670
N = 128 -6.610807 -6.611241 -6.610799 0.000067 -1.984670 0.000067 -1.964295 -6.610807 -6.610807 1.042794e-10 -3.999791 1.042794e-10 -5.591900
N = 256 -6.610807 -6.610916 -6.610806 0.000017 -1.992319 0.000017 -1.982094 -6.610807 -6.610807 2.507770e-09 -3.897269 2.507770e-09 -0.172920
N = 512 -6.610807 -6.610834 -6.610807 0.000004 -1.996154 0.000004 -1.991032 -6.610807 -6.610808 1.838673e-07 -3.969016 1.838673e-07 0.000192
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 4.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -8.118713 -8.460794 NaN 0.042135 NaN NaN NaN -8.133066 NaN 1.767875e-03 NaN NaN NaN
N = 8 -8.118713 -8.208090 NaN 0.011009 -1.936363 NaN NaN -8.119747 NaN 1.273164e-04 -3.795526 NaN NaN
N = 16 -8.118713 -8.141329 -8.117358 0.002953 -1.982599 0.002953 NaN -8.118781 -8.118705 9.315394e-06 -3.945395 9.315394e-06 NaN
N = 32 -8.118713 -8.124385 -8.118622 0.000710 -1.995544 0.000710 -1.978241 -8.118718 -8.118713 5.385828e-07 -3.986111 5.385828e-07 -3.942616
N = 64 -8.118713 -8.120132 -8.118708 0.000175 -1.998879 0.000175 -1.994429 -8.118714 -8.118713 3.291068e-08 -3.996505 3.291068e-08 -3.986725
N = 128 -8.118713 -8.119068 -8.118713 0.000044 -1.999719 0.000044 -1.998599 -8.118713 -8.118713 1.880380e-09 -3.999265 1.880380e-09 -4.111819
N = 256 -8.118713 -8.118802 -8.118713 0.000011 -1.999929 0.000011 -1.999650 -8.118713 -8.118713 6.711957e-09 -3.957658 6.711957e-09 -0.361555
N = 512 -8.118713 -8.118736 -8.118713 0.000003 -1.999969 0.000003 -1.999914 -8.118713 -8.118713 3.259909e-08 -4.511233 3.259909e-08 -0.005554
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 4.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -8.118713 -8.531236 NaN 0.050811 NaN NaN NaN -8.120861 NaN 2.645577e-04 NaN NaN NaN
N = 8 -8.118713 -8.240802 NaN 0.015038 -1.756541 NaN NaN -8.118855 NaN 1.741262e-05 -3.925377 NaN NaN
N = 16 -8.118713 -8.152065 -8.113024 0.004812 -1.872107 0.004812 NaN -8.118722 -8.118713 1.152191e-06 -3.979180 1.152191e-06 NaN
N = 32 -8.118713 -8.127440 -8.117983 0.001165 -1.934202 0.001165 -1.849443 -8.118714 -8.118713 7.010843e-08 -3.994197 7.010843e-08 -3.978225
N = 64 -8.118713 -8.120946 -8.118620 0.000287 -1.966597 0.000287 -1.922894 -8.118713 -8.118713 4.102870e-09 -3.998206 4.102870e-09 -4.072979
N = 128 -8.118713 -8.119278 -8.118702 0.000071 -1.983167 0.000071 -1.960943 -8.118713 -8.118713 3.686867e-09 -4.000771 3.686867e-09 3.285984
N = 256 -8.118713 -8.118855 -8.118712 0.000018 -1.991549 0.000018 -1.980339 -8.118713 -8.118713 3.290053e-08 -4.170778 3.290053e-08 -0.011173
N = 512 -8.118713 -8.118749 -8.118713 0.000004 -1.995766 0.000004 -1.990135 -8.118713 -8.118720 8.054074e-07 -5.371643 8.054074e-07 0.000028
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 8.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -9.254592 -9.698320 NaN 0.047947 NaN NaN NaN -9.274971 NaN 2.202047e-03 NaN NaN NaN
N = 8 -9.254592 -9.370579 NaN 0.012533 -1.935704 NaN NaN -9.256061 NaN 1.587889e-04 -3.793663 NaN NaN
N = 16 -9.254592 -9.283942 -9.252810 0.003365 -1.982521 0.003365 NaN -9.254687 -9.254579 1.163493e-05 -3.944816 1.163493e-05 NaN
N = 32 -9.254592 -9.261952 -9.254472 0.000808 -1.995531 0.000808 -1.978140 -9.254598 -9.254591 6.722891e-07 -3.985957 6.722891e-07 -3.942006
N = 64 -9.254592 -9.256433 -9.254584 0.000200 -1.998876 0.000200 -1.994413 -9.254592 -9.254592 4.113100e-08 -3.996495 4.113100e-08 -3.984620
N = 128 -9.254592 -9.255052 -9.254591 0.000050 -1.999719 0.000050 -1.998595 -9.254592 -9.254592 2.593194e-09 -3.998491 2.593194e-09 -3.976249
N = 256 -9.254592 -9.254707 -9.254592 0.000012 -1.999930 0.000012 -1.999649 -9.254592 -9.254592 9.260652e-10 -4.020792 9.260652e-10 -0.664331
N = 512 -9.254592 -9.254620 -9.254592 0.000003 -1.999984 0.000003 -1.999911 -9.254592 -9.254591 7.338935e-08 -2.728397 7.338935e-08 -0.003551
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 8.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -9.254592 -9.779991 NaN 0.056772 NaN NaN NaN -9.257242 NaN 2.863730e-04 NaN NaN NaN
N = 8 -9.254592 -9.409744 NaN 0.016765 -1.759727 NaN NaN -9.254767 NaN 1.891276e-05 -3.920464 NaN NaN
N = 16 -9.254592 -9.297012 -9.247662 0.005337 -1.870848 0.005337 NaN -9.254603 -9.254591 1.256283e-06 -3.978635 1.256283e-06 NaN
N = 32 -9.254592 -9.265704 -9.253665 0.001301 -1.932621 0.001301 -1.848271 -9.254592 -9.254592 7.620138e-08 -3.994470 7.620138e-08 -3.977814
N = 64 -9.254592 -9.257437 -9.254471 0.000321 -1.965528 0.000321 -1.921119 -9.254592 -9.254592 4.584467e-09 -3.998630 4.584467e-09 -4.034829
N = 128 -9.254592 -9.255312 -9.254576 0.000079 -1.982557 0.000079 -1.959714 -9.254592 -9.254592 3.057780e-10 -3.990863 3.057780e-10 -3.947723
N = 256 -9.254592 -9.254773 -9.254590 0.000020 -1.991225 0.000020 -1.979633 -9.254592 -9.254592 7.358738e-09 -4.171614 7.358738e-09 0.055911
N = 512 -9.254592 -9.254637 -9.254591 0.000005 -1.995597 0.000005 -1.989760 -9.254592 -9.254590 2.246012e-07 -2.898591 2.246012e-07 -0.000120
Convergence of FDM Heat with x_bar = 0.637 and k_ratio = 16.0
True Q FDM 2 Q FDM 2 Extrap Q FDM 2 % Error FDM 2 Exact B FDM 2 Extrap % Error FDM 2 Extrap B FDM 4 Q FDM 4 Extrap Q FDM 4 % Error FDM 4 Exact B FDM 4 Extrap % Error FDM 4 Extrap B
N = 4 -9.980639 -10.508396 NaN 0.052878 NaN NaN NaN -10.005565 NaN 2.497488e-03 NaN NaN NaN
N = 8 -9.980639 -10.118258 NaN 0.013789 -1.939192 NaN NaN -9.982436 NaN 1.800926e-04 -3.793667 NaN NaN
N = 16 -9.980639 -10.015440 -9.978646 0.003687 -1.983481 0.003687 NaN -9.980755 -9.980624 1.319577e-05 -3.944740 1.319577e-05 NaN
N = 32 -9.980639 -9.989364 -9.980505 0.000888 -1.995777 0.000888 -1.979342 -9.980646 -9.980638 7.625708e-07 -3.985932 7.625708e-07 -3.941925
N = 64 -9.980639 -9.982822 -9.980630 0.000220 -1.998938 0.000220 -1.994720 -9.980639 -9.980639 4.663319e-08 -3.996465 4.663319e-08 -3.985180
N = 128 -9.980639 -9.981185 -9.980638 0.000055 -1.999734 0.000055 -1.998673 -9.980639 -9.980639 3.043996e-09 -3.997001 3.043996e-09 -3.929875
N = 256 -9.980639 -9.980775 -9.980639 0.000014 -1.999931 0.000014 -1.999669 -9.980639 -9.980639 1.196629e-09 -4.015425 1.196629e-09 -0.345699
N = 512 -9.980639 -9.980673 -9.980639 0.000003 -1.999988 0.000003 -1.999911 -9.980639 -9.980638 8.575384e-08 -1.042866 8.575384e-08 -0.001550
Convergence of FEM Heat with x_bar = 0.637 and k_ratio = 16.0
True Q FEM 2 Q FEM 2 Extrap Q FEM 2 % Error FEM 2 Exact B FEM 2 Extrap % Error FEM 2 Extrap B FEM 4 Q FEM 4 Extrap Q FEM 4 % Error FEM 4 Exact B FEM 4 Extrap % Error FEM 4 Extrap B
N = 4 -9.980639 -10.599453 NaN 0.062001 NaN NaN NaN -9.983783 NaN 3.150905e-04 NaN NaN NaN
N = 8 -9.980639 -10.162539 NaN 0.018225 -1.766361 NaN NaN -9.980847 NaN 2.083072e-05 -3.918981 NaN NaN
N = 16 -9.980639 -10.030332 -9.972971 0.005752 -1.872008 0.005752 NaN -9.980652 -9.980638 1.385283e-06 -3.978393 1.385283e-06 NaN
N = 32 -9.980639 -9.993657 -9.979577 0.001411 -1.932506 0.001411 -1.849908 -9.980639 -9.980639 8.398012e-08 -3.994484 8.398012e-08 -3.977219
N = 64 -9.980639 -9.983973 -9.980498 0.000348 -1.965272 0.000348 -1.921052 -9.980639 -9.980639 5.199889e-09 -3.998709 5.199889e-09 -3.995296
N = 128 -9.980639 -9.981482 -9.980620 0.000086 -1.982375 0.000086 -1.959430 -9.980639 -9.980639 9.369049e-10 -4.010246 9.369049e-10 -2.067819
N = 256 -9.980639 -9.980851 -9.980636 0.000021 -1.991121 0.000021 -1.979424 -9.980639 -9.980638 2.628290e-08 -4.208162 2.628290e-08 -0.016616
N = 512 -9.980639 -9.980692 -9.980638 0.000005 -1.995533 0.000005 -1.989640 -9.980639 -9.980633 5.895397e-07 -3.717511 5.895397e-07 -0.000039
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