P2 moment fitting

The problems appear to be with boundary triangles that are very small compared to mesh size.

Old method with $h^{-1}$ subtriangles fitting the boundary

h        Nodes   Time      |   L2(Gamma)            H1(Gamma)    
---------------------------------------------------------------------------------------
0.1000      10    0:0:1s    |  1.44e-03                8.17e-02
0.0500      20    0:0:4s    |  1.93e-04   2.91      2.25e-02   1.86
0.0250      40   0:0:16s   |  2.56e-05   2.91      5.84e-03   1.95
0.0125      80    0:1:2s    |  3.28e-06   2.96      1.49e-03   1.97
0.0063     160   0:4:25s  |  4.14e-07   2.99      3.73e-04   2.00
0.0031     320  0:22:13s |  5.21e-08   2.99      9.38e-05   1.99
----------------------------------------------------------------------------------

Moments fitting P2
\begin{array}[h] {lrr|rr|rr} h & d.o.f & time & L_2(\Gamma) & & H_1(\Gamma) & \\ \hline 0.1000 &    10 &    0:0:1s &  2.58e-03  & & 8.16e-02 & \\0.05000 &    20 &    0:0:2s &  7.33e-04 & 1.82 & 2.29e-02 & 1.84 \\0.02500 &    40 &    0:0:4s &  1.47e-04 & 2.32 & 9.77e-03 & 1.23 \\0.01250 &    80 &    0:0:8s &  5.25e-05 & 1.49 & 7.82e-03 & 0.32 \\0.00625 &   160 &   0:0:15s &  8.28e-06 & 2.66 & 1.66e-03 & 2.24 \\0.00313 &   320 &   0:0:31s &  2.47e-06 & 1.74 & 1.05e-03 & 0.67 \\0.00156 &   640 &    0:1:2s &  1.82e-06 & 0.44 & 1.75e-03 & -0.74 \\0.00078 &  1280 &    0:2:6s &  2.34e-07 & 2.96 & 4.21e-04 & 2.06 \\0.00039 &  2560 &   0:4:14s &  2.81e-07 & -0.27 & 9.64e-04 & -1.20 \\0.00020 &  5120 &   0:8:42s &  2.47e-06 & -3.14 & 2.11e-02 & -4.45 \\0.00010 & 10240 &  0:19:22s &  1.82e-06 & 0.45 & 4.93e-02 & -1.22 \\\end{array}

P1 FEM with moments fitting integration on boundary

To figure out the problem with P2 MF scheme, I went back to P1 elements and use the moments fitting scheme of h^2 precision. The idea works, but there appears to be some numerical problem when I get to higher precision. The L2 error stagnates at 1E-8.

Update: this is not an error. We have seen this problem before - the strip width of 2h is just too small. 5h width has better results.

Strip width = 2h (surface $\pm h$)
Old method
\begin{array}[h] {lrr|rr|rr} h & d.o.f & time & L_2(\Gamma) & & H_1(\Gamma) & \\ \hline 0.1000 &    10 &    0:0:0s &  5.50e-02  & & 1.23e+00 & \\0.05000 &    20 &    0:0:1s &  1.55e-02 & 1.83 & 6.19e-01 & 0.99 \\0.02500 &    40 &    0:0:1s &  4.28e-03 & 1.86 & 3.19e-01 & 0.96 \\0.01250 &    80 &    0:0:2s &  1.08e-03 & 1.99 & 1.60e-01 & 0.99 \\0.00625 &   160 &    0:0:3s &  2.64e-04 & 2.02 & 7.98e-02 & 1.00 \\0.00313 &   320 &    0:0:6s &  6.68e-05 & 1.98 & 3.99e-02 & 1.00 \\0.00156 &   640 &   0:0:13s &  1.59e-05 & 2.07 & 1.99e-02 & 1.00 \\0.00078 &  1280 &   0:0:26s &  4.22e-06 & 1.92 & 9.90e-03 & 1.01 \\0.00039 &  2560 &   0:0:50s &  1.36e-06 & 1.63 & 4.91e-03 & 1.01 \\0.00020 &  5120 &   0:1:40s &  3.10e-07 & 2.14 & 2.41e-03 & 1.03 \\0.00010 & 10240 &   0:3:22s &  8.48e-08 & 1.87 & 1.16e-03 & 1.05 \\0.00005 & 20480 &   0:6:51s &  3.69e-08 & 1.20 & 5.35e-04 & 1.12 \\0.00002 & 40960 &   0:14:6s &  8.42e-09 & 2.13 & 2.23e-04 & 1.26 \\\end{array}

Moments fitting Strip width = 2h (surface $\pm h$)
\begin{array}[h] {lrr|rr|rr} h & d.o.f & time & L_2(\Gamma) & & H_1(\Gamma) & \\ \hline 0.1000 &    10 &   0:0:1s &  5.74e-02  & & 1.26e+00 & \\0.05000 &    20 &    0:0:1s &  1.40e-02 & 2.04 & 6.29e-01 & 1.01 \\0.02500 &    40 &    0:0:2s &  3.68e-03 & 1.92 & 3.20e-01 & 0.97 \\0.01250 &    80 &    0:0:3s &  8.99e-04 & 2.03 & 1.60e-01 & 1.00 \\0.00625 &   160 &    0:0:6s &  2.39e-04 & 1.91 & 7.98e-02 & 1.00 \\0.00313 &   320 &   0:0:11s &  5.82e-05 & 2.04 & 3.99e-02 & 1.00 \\0.00156 &   640 &   0:0:23s &  1.65e-05 & 1.82 & 1.99e-02 & 1.00 \\0.00078 &  1280 &   0:0:45s &  4.83e-06 & 1.77 & 9.90e-03 & 1.01 \\0.00039 &  2560 &   0:1:30s &  1.24e-06 & 1.96 & 4.91e-03 & 1.01 \\0.00020 &  5120 &    0:3:5s &  4.10e-07 & 1.60 & 2.41e-03 & 1.03 \\0.00010 & 10240 &   0:6:40s &  1.28e-07 & 1.68 & 1.16e-03 & 1.05 \\0.00005 & 20480 &   0:15:6s &  4.65e-08 & 1.46 & 5.35e-04 & 1.12 \\0.00002 & 40960 &  0:36:21s &  1.49e-08 & 1.64 & 2.23e-04 & 1.26 \\0.00001 & 81920 &  1:37:48s &  2.78e-08 & -0.90 & 6.74e-05 & 1.72 \\0.00001 & 163840 &   4:54:3s &  3.20e-08 & -0.20 & 1.25e-05 & 2.43 \\\end{array}

Old method Strip width = 5h (surface $\pm 2.5 h$)
\begin{array}[h] {lrr|rr|rr} h & d.o.f & time & L_2(\Gamma) & & H_1(\Gamma) & \\ \hline 0.1000 &    10 &    0:0:3s &  6.19e-02  & & 1.27e+00 & \\0.05000 &    20 &    0:0:4s &  1.51e-02 & 2.03 & 6.29e-01 & 1.01 \\0.02500 &    40 &    0:0:6s &  3.78e-03 & 2.00 & 3.20e-01 & 0.97 \\0.01250 &    80 &    0:0:5s &  9.32e-04 & 2.02 & 1.60e-01 & 1.00 \\0.00625 &   160 &    0:0:7s &  2.34e-04 & 1.99 & 7.99e-02 & 1.00 \\0.00313 &   320 &   0:0:12s &  5.88e-05 & 2.00 & 3.99e-02 & 1.00 \\0.00156 &   640 &   0:0:21s &  1.46e-05 & 2.01 & 2.00e-02 & 1.00 \\0.00078 &  1280 &   0:0:36s &  3.65e-06 & 2.00 & 9.98e-03 & 1.00 \\0.00039 &  2560 &    0:1:6s &  8.82e-07 & 2.05 & 4.99e-03 & 1.00 \\0.00020 &  5120 &    0:2:8s &  2.34e-07 & 1.91 & 2.49e-03 & 1.00 \\0.00010 & 10240 &   0:4:15s &  5.41e-08 & 2.11 & 1.24e-03 & 1.01 \\0.00005 & 20480 &   0:8:20s &  1.34e-08 & 2.02 & 6.16e-04 & 1.01 \\\end{array}

Moments fitting Strip width = 5h (surface $\pm 2.5 h$)
\begin{array}[h] {lrr|rr|rr} h & d.o.f & time & L_2(\Gamma) & & H_1(\Gamma) & \\ \hline 0.1000 &    10 &    0:0:4s &  6.21e-02  & & 1.27e+00 & \\0.05000 &    20 &    0:0:5s &  1.51e-02 & 2.04 & 6.30e-01 & 1.01 \\0.02500 &    40 &    0:0:8s &  3.73e-03 & 2.02 & 3.20e-01 & 0.98 \\0.01250 &    80 &    0:0:8s &  9.34e-04 & 2.00 & 1.60e-01 & 1.00 \\0.00625 &   160 &   0:0:13s &  2.28e-04 & 2.03 & 7.99e-02 & 1.00 \\0.00313 &   320 &   0:0:22s &  5.74e-05 & 1.99 & 3.99e-02 & 1.00 \\0.00156 &   640 &   0:0:40s &  1.44e-05 & 2.00 & 2.00e-02 & 1.00 \\0.00078 &  1280 &   0:1:15s &  3.61e-06 & 1.99 & 9.98e-03 & 1.00 \\0.00039 &  2560 &   0:2:28s &  9.15e-07 & 1.98 & 4.99e-03 & 1.00 \\0.00020 &  5120 &   0:4:51s &  2.26e-07 & 2.02 & 2.49e-03 & 1.00 \\0.00010 & 10240 &  0:10:18s &  5.62e-08 & 2.01 & 1.24e-03 & 1.01 \\0.00005 & 20480 &  0:22:33s &  1.43e-08 & 1.97 & 6.16e-04 & 1.01 \\\end{array}

Moments fitting, H=0
\begin{array}[h] {lrr|rr|rr} h & d.o.f & time & L_2(\Gamma) & & H_1(\Gamma) & \\ \hline 0.1000 &    10 &    0:0:1s &  5.27e-02  & & 1.23e+00 & \\0.0500 &    20 &    0:0:1s &  1.46e-02 & 1.85 & 6.19e-01 & 0.99 \\0.0250 &    40 &    0:0:2s &  4.11e-03 & 1.83 & 3.19e-01 & 0.96 \\0.0125 &    80 &    0:0:3s &  1.02e-03 & 2.01 & 1.60e-01 & 1.00 \\0.0063 &   160 &    0:0:5s &  2.69e-04 & 1.92 & 7.97e-02 & 1.00 \\0.0031 &   320 &   0:0:11s &  6.59e-05 & 2.03 & 3.99e-02 & 1.00 \\0.0016 &   640 &   0:0:21s &  1.84e-05 & 1.84 & 1.99e-02 & 1.00 \\0.0008 &  1280 &   0:0:43s &  5.21e-06 & 1.82 & 9.90e-03 & 1.01 \\0.0004 &  2560 &   0:1:25s &  1.32e-06 & 1.98 & 4.91e-03 & 1.01 \\0.0002 &  5120 &   0:2:58s &  4.30e-07 & 1.61 & 2.41e-03 & 1.03 \\\end{array}