Applying previous method to the Laplace equation on the unit circle with Monte-Carlo integration, I got following results:
h Nodes Time | L2(Gamma) H1(Gamma) Area*St.dev
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0.1000 353 0:0:1s | 1.71e-03 8.15e-02 3.01e-03
0.0500 697 0:0:1s | 3.14e-04 2.45 2.25e-02 1.86 5.58e-04 2.43
0.0250 1361 0:0:4s | 3.39e-05 3.21 5.84e-03 1.95 9.86e-05 2.50
0.0125 2717 0:0:19s | 3.29e-06 3.37 1.49e-03 1.97 1.66e-05 2.57
0.0063 5457 0:5:16s | 4.14e-07 2.99 3.73e-04 2.00 3.07e-06 2.44
0.0031 10837 2:46:24s | 5.21e-08 2.99 9.38e-05 1.99 4.37e-07 2.81
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Another way to integrate irregular areas in unfitted triangles is to place ${h^-1}$ points on the domain boundary, triangulate and integrate.
h Nodes Time | L2(Gamma) H1(Gamma) Area*St.dev
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0.1000 353 0:0:1s | 1.43e-03 8.14e-02 0.00e+00
0.0500 697 0:0:4s | 1.93e-04 2.89 2.25e-02 1.86 0.00e+00 NaN
0.0250 1361 0:0:17s | 2.55e-05 2.91 5.83e-03 1.94 0.00e+00 NaN
0.0125 2717 0:1:5s | 3.28e-06 2.96 1.49e-03 1.97 0.00e+00 NaN
0.0063 5457 0:4:53s | 4.14e-07 2.99 3.73e-04 2.00 0.00e+00 NaN
0.0031 10837 0:26:53s | 5.21e-08 2.99 9.38e-05 1.99 0.00e+00 NaN
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I am surprised to get very similar results for the Monte-Carlo and triangulation method. This seems a bit suspicious.
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