This page was generated from wta/poisson.ipynb.
Poisson EquationΒΆ
Import Netgen/NGSolve Python modules:
[1]:
from ngsolve import *
from ngsolve.webgui import Draw
The unit_square is a predefined domain, use Netgen to generate a mesh:
[2]:
unit_square.GenerateMesh(maxh=0.2)
[2]:
<netgen.libngpy._meshing.Mesh at 0x7fc996c77e70>
[3]:
mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))
Draw (mesh);
Define a finite element space on that mesh.
[4]:
fes = H1(mesh, order=3, dirichlet="left|right|bottom|top")
print ("ndof =", fes.ndof)
ndof = 274
[5]:
print (mesh.GetBoundaries())
('bottom', 'right', 'top', 'left')
Define linear and bilinear-forms.
\[a(u,v) = \int \nabla u \nabla v
\qquad \text{and} \qquad
f(v) = \int f v\]
Forms are expressed in terms of trial and test-functions:
[6]:
u = fes.TrialFunction()
v = fes.TestFunction()
f = LinearForm(fes)
f += 32*(y*(1-y)+x*(1-x))*v*dx
a = BilinearForm(fes)
a += grad(u)*grad(v)*dx
a.Assemble()
f.Assemble();
[7]:
print(f.vec)
print(a.mat)
0.0196267
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Row 22: 6: -0.592403 7: -0.892641 8: -0.247724 21: -0.455722 22: 3.65035 23: -0.897438 32: -0.56442 68: 0.0934201 69: -1.21431e-17 70: 0.099944 71: -6.93889e-18 72: -0.0946304 73: 3.66027e-16 74: 0.124774 75: 7.28584e-17 76: -0.0694209 77: 5.3603e-16 80: -0.188936 81: 4.21538e-16 82: 0.105449 83: -9.36751e-17 148: -0.106594 149: 3.10949e-16 150: 0.0826032 151: -4.51028e-17 154: -0.0580765 155: -4.36283e-16 156: -0.0907341 157: -3.56486e-16 160: 0.102201 161: -1.21431e-17 224: 6.59195e-17 225: -3.98986e-17 227: 8.67362e-17 253: -3.64292e-17 258: 6.85216e-17 259: -2.77556e-17
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Row 161: 22: -1.21431e-17 23: -2.9924e-16 32: 3.747e-16 36: -5.55112e-17 154: 0.00511004 155: 0.00149193 156: -0.00511004 157: -0.00191477 160: -0.000238659 161: 0.0219102 162: 0.00643218 163: -0.00243503 204: -0.00643218 205: 0.00185309 259: -0.000422836 268: 0.000581942
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Row 165: 10: 8.67362e-17 24: -3.88144e-16 25: 3.17021e-16 34: -1.73472e-17 92: 0.00513751 93: 0.00101359 94: -0.00513751 95: -0.00241142 164: 0.0010385 165: 0.0226634 166: 0.0050486 167: -0.00203561 172: -0.0050486 173: 0.00133012 234: -0.00139783 256: 0.000705495
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Row 167: 24: -2.88831e-16 25: -4.85723e-17 34: 3.27429e-16 36: 1.38778e-17 164: 0.00528552 165: -0.00203561 166: -0.000321294 167: 0.0218159 168: 0.00640004 169: -0.00196667 172: -0.00528552 173: -0.00148807 212: -0.00640004 213: 0.00230002 256: 0.000547548 270: -0.000333352
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Row 170: 12: 0.07774 25: -0.0534755 26: -0.124305 34: 0.100041 102: -0.0182617 103: -0.00789829 104: -0.00117332 105: -0.00448062 170: 0.0767803 171: -0.00354149 172: -0.0128147 173: 0.00478399 176: -0.0121955 177: 0.00466017 251: -0.00849191 257: -0.00444213
Row 171: 12: -6.67869e-17 25: -3.8294e-16 26: 4.55365e-16 34: -8.67362e-19 102: 0.003887 103: 0.00243489 104: -0.003887 105: -0.000156443 170: -0.00354149 171: 0.0247861 172: 0.00500204 173: -0.00170862 176: -0.00500204 177: 0.00162607 251: 0.00227845 257: 8.25482e-05
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Row 173: 24: -1.9082e-17 25: -3.95517e-16 26: -4.16334e-17 34: 4.46258e-16 164: 0.00422728 165: 0.00133012 166: -0.00422728 167: -0.00148807 170: 0.00478399 171: -0.00170862 172: -0.000104947 173: 0.0230001 176: -0.00478399 177: -0.00148071 256: -0.000157948 257: 0.000227912
Row 174: 13: 0.0894845 26: -0.138872 27: -0.0444174 34: 0.093805 108: -0.00417457 109: -0.00403605 110: -0.0181966 111: -0.00684045 174: 0.0754537 175: 0.00472274 176: -0.00692977 177: 0.00411108 180: -0.0165215 181: 0.00602942 238: -0.00640227 263: -0.00545025
Row 175: 13: 1.02349e-16 26: -4.81386e-16 27: 3.39572e-16 34: 3.72966e-17 108: 0.00447423 109: 0.000556609 110: -0.00447423 111: -0.00242621 174: 0.00472274 175: 0.0240718 176: 0.00469025 177: -0.000923969 180: -0.00469025 181: 0.00220286 238: -0.0018696 263: -0.0012789
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Row 177: 25: -3.29597e-17 26: -4.25007e-16 27: 1.38778e-17 34: 4.45824e-16 170: 0.00466017 171: 0.00162607 172: -0.00466017 173: -0.00148071 174: 0.00411108 175: -0.000923969 176: 0.00112112 177: 0.0234295 180: -0.00411108 181: -0.00181675 257: 0.000145364 263: -0.000892779
Row 178: 14: 0.0896011 27: -0.0642451 28: -0.114524 35: 0.089168 114: -0.0155802 115: -0.00604253 116: -0.00682004 117: -0.00429049 178: 0.0733846 179: -0.00251394 182: -0.0130507 183: 0.00542212 190: -0.00924125 191: 0.00466022 240: -0.00585296 262: -0.00562395
Row 179: 14: -1.9082e-17 27: -3.50414e-16 28: 3.89445e-16 35: -1.73472e-17 114: 0.00448006 115: 0.00207737 116: -0.00448006 117: -0.000909338 178: -0.00251394 179: 0.0233948 182: 0.0044584 183: -0.0017401 190: -0.0044584 191: 0.00123217 240: 0.00116803 262: 0.000507933
Row 180: 26: 0.120588 27: -0.0838915 34: -0.139262 37: 0.102565 174: -0.0165215 175: -0.00469025 176: -0.0136256 177: -0.00411108 180: 0.0744031 181: -0.00276852 184: -0.018294 185: 0.00599579 214: -0.00734725 215: 0.00380644 228: -0.00467399 263: -0.00277191
Row 181: 26: -5.20417e-17 27: -2.94252e-16 34: 4.31946e-16 37: -9.02056e-17 174: 0.00602942 175: 0.00220286 176: -0.00602942 177: -0.00181675 180: -0.00276852 181: 0.0223228 184: 0.00512825 185: -0.0024392 214: -0.00512825 215: 0.000979634 228: 0.00145957 263: 0.000386116
Row 182: 27: -0.0900435 28: 0.108442 35: -0.123241 37: 0.104842 178: -0.0130507 179: 0.0044584 182: 0.0733955 183: -0.0016599 184: -0.0177596 185: 0.00571854 190: -0.0140599 191: -0.00466022 216: -0.008451 217: 0.00385682 262: -0.0036965 269: -0.00433323
Row 183: 27: -2.95337e-16 28: -2.08167e-17 35: 3.52149e-16 37: -4.25007e-17 178: 0.00542212 179: -0.0017401 182: -0.0016599 183: 0.0222487 184: 0.00524212 185: -0.00236795 190: -0.00542212 191: -0.00187465 216: -0.00524212 217: 0.0011268 262: -0.000134549 269: 0.00124115
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Row 185: 27: -2.46331e-16 34: -2.94903e-17 35: -7.28584e-17 37: 3.49547e-16 180: 0.00599579 181: -0.0024392 182: 0.00571854 183: -0.00236795 184: -0.00270711 185: 0.0219384 214: -0.00599579 215: -0.001558 216: -0.00571854 217: -0.00144441 228: 0.000881204 269: 0.000923535
Row 186: 15: 0.0967187 28: -0.053174 29: -0.132079 30: 0.0885341 118: -0.0191557 119: -0.00662515 120: -0.00502395 121: -0.00379879 186: 0.0747129 187: -0.00394524 188: -0.013864 189: 0.00565872 192: -0.00826956 193: 0.00453984 242: -0.00558801 260: -0.00577185
Row 187: 15: -4.68375e-17 28: -3.50414e-16 29: 4.17635e-16 30: -2.1684e-17 118: 0.00483593 119: 0.0025541 120: -0.00483593 121: -0.00066986 186: -0.00394524 187: 0.02371 188: 0.00442671 189: -0.00184853 192: -0.00442671 193: 0.00110261 242: 0.00188424 260: 0.000745922
Row 188: 28: -0.0970242 29: 0.113174 30: -0.107391 35: 0.0912412 186: -0.013864 187: 0.00442671 188: 0.0728079 189: -0.000518359 190: -0.0129839 191: 0.00528366 192: -0.0144296 193: -0.00453984 196: -0.00982642 197: 0.00465217 260: -0.00330782 261: -0.00537376
Row 189: 28: -3.08781e-16 29: -2.77556e-17 30: 3.53233e-16 35: -2.08167e-17 186: 0.00565872 187: -0.00184853 188: -0.000518359 189: 0.0223093 190: 0.00456206 191: -0.00173118 192: -0.00565872 193: -0.00192395 196: -0.00456206 197: 0.00131019 260: -7.54207e-05 261: 0.000420993
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Row 191: 27: 0 28: -3.66027e-16 30: -2.77556e-17 35: 3.97685e-16 178: 0.00466022 179: 0.00123217 182: -0.00466022 183: -0.00187465 188: 0.00528366 189: -0.00173118 190: 0.00105383 191: 0.0224094 196: -0.00528366 197: -0.00179125 261: -6.00706e-05 262: -0.000642483
Row 192: 16: 0.100924 28: 0.0907968 29: -0.134718 30: -0.0570021 124: -0.00598097 125: -0.003724 126: -0.01925 127: -0.0063778 186: -0.00826956 187: -0.00442671 188: -0.0144296 189: -0.00565872 192: 0.0744331 193: 0.00388582 245: -0.00505561 260: -0.00554559
Row 193: 16: 1.00614e-16 28: 6.93889e-18 29: -4.7358e-16 30: 3.71231e-16 124: 0.00504619 125: 0.000797462 126: -0.00504619 127: -0.00256666 186: 0.00453984 187: 0.00110261 188: -0.00453984 189: -0.00192395 192: 0.00388582 193: 0.0233826 245: -0.0017692 260: -0.000821343
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Row 195: 17: 6.59195e-17 30: -3.97252e-16 31: 3.16587e-16 35: 1.21431e-17 130: 0.00495486 131: 0.00116329 132: -0.00495486 133: -0.00213995 194: 0.00203981 195: 0.0225197 196: 0.00496121 197: -0.00146212 200: -0.00496121 201: 0.00184535 247: -0.000976655 264: -0.00038322
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Row 197: 28: -1.04083e-17 30: -4.05925e-16 31: -2.25514e-17 35: 4.34982e-16 188: 0.00465217 189: 0.00131019 190: -0.00465217 191: -0.00179125 194: 0.00446891 195: -0.00146212 196: 0.000804132 197: 0.022891 200: -0.00446891 201: -0.00151715 261: -0.000481064 264: -5.50237e-05
Row 198: 20: 0.0869532 31: -0.0875396 33: -0.0977775 35: 0.0983639 144: -0.0140519 145: -0.00580986 146: -0.00768643 147: -0.00453678 198: 0.0729761 199: -0.000511899 200: -0.0103925 201: 0.00440838 208: -0.0141985 209: 0.00516957 265: -0.00599898 266: -0.00465976
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Solve the problem:
[8]:
gfu = GridFunction(fes)
gfu.vec.data = a.mat.Inverse(fes.FreeDofs()) * f.vec
Plot the solution:
[9]:
Draw (gfu, mesh);
[10]:
Draw (-grad(gfu), mesh, "Flux", vectors= { "grid_size" : 40});
Calculate error:
[11]:
exact = 16*x*(1-x)*y*(1-y)
print ("L2-error:", sqrt(Integrate((gfu-exact)**2, mesh)))
L2-error: 5.8102129540773155e-05
[ ]: