History of convergence of the HDG method for |$\delta =0$| and with 1 hanging node per interface edge
| |$k$| . | |$N$| . | #d.o.f. . | |$e(\mathrm{L})$| . | |$r$| . | |$e(\mathbf{u})$| . | |$r$| . | |$e(p)$| . | |$r$| . | |$\widehat{e}$| . | |$\widehat{r}$| . |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 34 | 208 | 2.81e-01 | * | 3.43e-01 | * | 1.84e-01 | * | 2.67e-02 | * |
| 134 | 812 | 6.62e-02 | 2.11 | 8.17e-02 | 2.09 | 4.52e-02 | 2.05 | 3.33e-03 | 3.03 | |
| 502 | 3060 | 1.74e-02 | 2.03 | 2.02e-02 | 2.11 | 1.17e-02 | 2.05 | 3.66e-04 | 3.34 | |
| 1972 | 12158 | 4.38e-03 | 2.01 | 5.18e-03 | 1.99 | 3.08e-03 | 1.95 | 5.14e-05 | 2.87 | |
| 7976 | 49410 | 1.09e-03 | 2.00 | 1.29e-03 | 1.98 | 7.70e-04 | 1.98 | 6.26e-06 | 3.01 | |
| 2 | 34 | 297 | 2.96e-02 | * | 7.25e-02 | * | 3.40e-02 | * | 2.53e-03 | * |
| 134 | 1164 | 3.66e-03 | 3.05 | 7.40e-03 | 3.33 | 3.67e-03 | 3.25 | 1.19e-04 | 4.45 | |
| 502 | 4391 | 5.65e-04 | 2.83 | 9.23e-04 | 3.15 | 5.03e-04 | 3.01 | 6.91e-06 | 4.31 | |
| 1972 | 17448 | 6.82e-05 | 3.09 | 1.39e-04 | 2.77 | 6.97e-05 | 2.89 | 5.62e-07 | 3.67 | |
| 7976 | 70913 | 8.43e-06 | 2.99 | 1.69e-05 | 3.01 | 8.63e-06 | 2.99 | 3.39e-08 | 4.02 | |
| 3 | 34 | 386 | 4.89e-03 | * | 1.50e-02 | * | 6.65e-03 | * | 2.48e-04 | * |
| 134 | 1516 | 2.29e-04 | 4.46 | 1.12e-03 | 3.78 | 3.89e-04 | 4.14 | 8.95e-06 | 4.84 | |
| 502 | 5722 | 1.56e-05 | 4.06 | 5.38e-05 | 4.60 | 2.21e-05 | 4.35 | 2.11e-07 | 5.68 | |
| 1972 | 22738 | 9.55e-07 | 4.09 | 3.64e-06 | 3.94 | 1.59e-06 | 3.85 | 7.42e-09 | 4.89 | |
| 7976 | 92416 | 6.00e-08 | 3.96 | 2.23e-07 | 3.99 | 9.81e-08 | 3.98 | 2.27e-10 | 4.99 | |
| 4 | 34 | 475 | 2.57e-04 | * | 2.65e-03 | * | 1.10e-03 | * | 2.30e-05 | * |
| 134 | 1868 | 8.65e-06 | 4.95 | 6.04e-05 | 5.51 | 3.31e-05 | 5.12 | 2.65e-07 | 6.51 | |
| 502 | 7053 | 3.97e-07 | 4.66 | 2.01e-06 | 5.15 | 9.66e-07 | 5.35 | 4.07e-09 | 6.32 | |
| 1972 | 28028 | 1.18e-08 | 5.14 | 9.16e-08 | 4.52 | 3.73e-08 | 4.76 | 9.82e-11 | 5.44 | |
| 7976 | 113919 | 3.68e-10 | 4.96 | 2.79e-09 | 5.00 | 1.14e-09 | 4.99 | 2.20e-12 | 5.44 |
| |$k$| . | |$N$| . | #d.o.f. . | |$e(\mathrm{L})$| . | |$r$| . | |$e(\mathbf{u})$| . | |$r$| . | |$e(p)$| . | |$r$| . | |$\widehat{e}$| . | |$\widehat{r}$| . |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 34 | 208 | 2.81e-01 | * | 3.43e-01 | * | 1.84e-01 | * | 2.67e-02 | * |
| 134 | 812 | 6.62e-02 | 2.11 | 8.17e-02 | 2.09 | 4.52e-02 | 2.05 | 3.33e-03 | 3.03 | |
| 502 | 3060 | 1.74e-02 | 2.03 | 2.02e-02 | 2.11 | 1.17e-02 | 2.05 | 3.66e-04 | 3.34 | |
| 1972 | 12158 | 4.38e-03 | 2.01 | 5.18e-03 | 1.99 | 3.08e-03 | 1.95 | 5.14e-05 | 2.87 | |
| 7976 | 49410 | 1.09e-03 | 2.00 | 1.29e-03 | 1.98 | 7.70e-04 | 1.98 | 6.26e-06 | 3.01 | |
| 2 | 34 | 297 | 2.96e-02 | * | 7.25e-02 | * | 3.40e-02 | * | 2.53e-03 | * |
| 134 | 1164 | 3.66e-03 | 3.05 | 7.40e-03 | 3.33 | 3.67e-03 | 3.25 | 1.19e-04 | 4.45 | |
| 502 | 4391 | 5.65e-04 | 2.83 | 9.23e-04 | 3.15 | 5.03e-04 | 3.01 | 6.91e-06 | 4.31 | |
| 1972 | 17448 | 6.82e-05 | 3.09 | 1.39e-04 | 2.77 | 6.97e-05 | 2.89 | 5.62e-07 | 3.67 | |
| 7976 | 70913 | 8.43e-06 | 2.99 | 1.69e-05 | 3.01 | 8.63e-06 | 2.99 | 3.39e-08 | 4.02 | |
| 3 | 34 | 386 | 4.89e-03 | * | 1.50e-02 | * | 6.65e-03 | * | 2.48e-04 | * |
| 134 | 1516 | 2.29e-04 | 4.46 | 1.12e-03 | 3.78 | 3.89e-04 | 4.14 | 8.95e-06 | 4.84 | |
| 502 | 5722 | 1.56e-05 | 4.06 | 5.38e-05 | 4.60 | 2.21e-05 | 4.35 | 2.11e-07 | 5.68 | |
| 1972 | 22738 | 9.55e-07 | 4.09 | 3.64e-06 | 3.94 | 1.59e-06 | 3.85 | 7.42e-09 | 4.89 | |
| 7976 | 92416 | 6.00e-08 | 3.96 | 2.23e-07 | 3.99 | 9.81e-08 | 3.98 | 2.27e-10 | 4.99 | |
| 4 | 34 | 475 | 2.57e-04 | * | 2.65e-03 | * | 1.10e-03 | * | 2.30e-05 | * |
| 134 | 1868 | 8.65e-06 | 4.95 | 6.04e-05 | 5.51 | 3.31e-05 | 5.12 | 2.65e-07 | 6.51 | |
| 502 | 7053 | 3.97e-07 | 4.66 | 2.01e-06 | 5.15 | 9.66e-07 | 5.35 | 4.07e-09 | 6.32 | |
| 1972 | 28028 | 1.18e-08 | 5.14 | 9.16e-08 | 4.52 | 3.73e-08 | 4.76 | 9.82e-11 | 5.44 | |
| 7976 | 113919 | 3.68e-10 | 4.96 | 2.79e-09 | 5.00 | 1.14e-09 | 4.99 | 2.20e-12 | 5.44 |
History of convergence of the HDG method for |$\delta =0$| and with 1 hanging node per interface edge
| |$k$| . | |$N$| . | #d.o.f. . | |$e(\mathrm{L})$| . | |$r$| . | |$e(\mathbf{u})$| . | |$r$| . | |$e(p)$| . | |$r$| . | |$\widehat{e}$| . | |$\widehat{r}$| . |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 34 | 208 | 2.81e-01 | * | 3.43e-01 | * | 1.84e-01 | * | 2.67e-02 | * |
| 134 | 812 | 6.62e-02 | 2.11 | 8.17e-02 | 2.09 | 4.52e-02 | 2.05 | 3.33e-03 | 3.03 | |
| 502 | 3060 | 1.74e-02 | 2.03 | 2.02e-02 | 2.11 | 1.17e-02 | 2.05 | 3.66e-04 | 3.34 | |
| 1972 | 12158 | 4.38e-03 | 2.01 | 5.18e-03 | 1.99 | 3.08e-03 | 1.95 | 5.14e-05 | 2.87 | |
| 7976 | 49410 | 1.09e-03 | 2.00 | 1.29e-03 | 1.98 | 7.70e-04 | 1.98 | 6.26e-06 | 3.01 | |
| 2 | 34 | 297 | 2.96e-02 | * | 7.25e-02 | * | 3.40e-02 | * | 2.53e-03 | * |
| 134 | 1164 | 3.66e-03 | 3.05 | 7.40e-03 | 3.33 | 3.67e-03 | 3.25 | 1.19e-04 | 4.45 | |
| 502 | 4391 | 5.65e-04 | 2.83 | 9.23e-04 | 3.15 | 5.03e-04 | 3.01 | 6.91e-06 | 4.31 | |
| 1972 | 17448 | 6.82e-05 | 3.09 | 1.39e-04 | 2.77 | 6.97e-05 | 2.89 | 5.62e-07 | 3.67 | |
| 7976 | 70913 | 8.43e-06 | 2.99 | 1.69e-05 | 3.01 | 8.63e-06 | 2.99 | 3.39e-08 | 4.02 | |
| 3 | 34 | 386 | 4.89e-03 | * | 1.50e-02 | * | 6.65e-03 | * | 2.48e-04 | * |
| 134 | 1516 | 2.29e-04 | 4.46 | 1.12e-03 | 3.78 | 3.89e-04 | 4.14 | 8.95e-06 | 4.84 | |
| 502 | 5722 | 1.56e-05 | 4.06 | 5.38e-05 | 4.60 | 2.21e-05 | 4.35 | 2.11e-07 | 5.68 | |
| 1972 | 22738 | 9.55e-07 | 4.09 | 3.64e-06 | 3.94 | 1.59e-06 | 3.85 | 7.42e-09 | 4.89 | |
| 7976 | 92416 | 6.00e-08 | 3.96 | 2.23e-07 | 3.99 | 9.81e-08 | 3.98 | 2.27e-10 | 4.99 | |
| 4 | 34 | 475 | 2.57e-04 | * | 2.65e-03 | * | 1.10e-03 | * | 2.30e-05 | * |
| 134 | 1868 | 8.65e-06 | 4.95 | 6.04e-05 | 5.51 | 3.31e-05 | 5.12 | 2.65e-07 | 6.51 | |
| 502 | 7053 | 3.97e-07 | 4.66 | 2.01e-06 | 5.15 | 9.66e-07 | 5.35 | 4.07e-09 | 6.32 | |
| 1972 | 28028 | 1.18e-08 | 5.14 | 9.16e-08 | 4.52 | 3.73e-08 | 4.76 | 9.82e-11 | 5.44 | |
| 7976 | 113919 | 3.68e-10 | 4.96 | 2.79e-09 | 5.00 | 1.14e-09 | 4.99 | 2.20e-12 | 5.44 |
| |$k$| . | |$N$| . | #d.o.f. . | |$e(\mathrm{L})$| . | |$r$| . | |$e(\mathbf{u})$| . | |$r$| . | |$e(p)$| . | |$r$| . | |$\widehat{e}$| . | |$\widehat{r}$| . |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 34 | 208 | 2.81e-01 | * | 3.43e-01 | * | 1.84e-01 | * | 2.67e-02 | * |
| 134 | 812 | 6.62e-02 | 2.11 | 8.17e-02 | 2.09 | 4.52e-02 | 2.05 | 3.33e-03 | 3.03 | |
| 502 | 3060 | 1.74e-02 | 2.03 | 2.02e-02 | 2.11 | 1.17e-02 | 2.05 | 3.66e-04 | 3.34 | |
| 1972 | 12158 | 4.38e-03 | 2.01 | 5.18e-03 | 1.99 | 3.08e-03 | 1.95 | 5.14e-05 | 2.87 | |
| 7976 | 49410 | 1.09e-03 | 2.00 | 1.29e-03 | 1.98 | 7.70e-04 | 1.98 | 6.26e-06 | 3.01 | |
| 2 | 34 | 297 | 2.96e-02 | * | 7.25e-02 | * | 3.40e-02 | * | 2.53e-03 | * |
| 134 | 1164 | 3.66e-03 | 3.05 | 7.40e-03 | 3.33 | 3.67e-03 | 3.25 | 1.19e-04 | 4.45 | |
| 502 | 4391 | 5.65e-04 | 2.83 | 9.23e-04 | 3.15 | 5.03e-04 | 3.01 | 6.91e-06 | 4.31 | |
| 1972 | 17448 | 6.82e-05 | 3.09 | 1.39e-04 | 2.77 | 6.97e-05 | 2.89 | 5.62e-07 | 3.67 | |
| 7976 | 70913 | 8.43e-06 | 2.99 | 1.69e-05 | 3.01 | 8.63e-06 | 2.99 | 3.39e-08 | 4.02 | |
| 3 | 34 | 386 | 4.89e-03 | * | 1.50e-02 | * | 6.65e-03 | * | 2.48e-04 | * |
| 134 | 1516 | 2.29e-04 | 4.46 | 1.12e-03 | 3.78 | 3.89e-04 | 4.14 | 8.95e-06 | 4.84 | |
| 502 | 5722 | 1.56e-05 | 4.06 | 5.38e-05 | 4.60 | 2.21e-05 | 4.35 | 2.11e-07 | 5.68 | |
| 1972 | 22738 | 9.55e-07 | 4.09 | 3.64e-06 | 3.94 | 1.59e-06 | 3.85 | 7.42e-09 | 4.89 | |
| 7976 | 92416 | 6.00e-08 | 3.96 | 2.23e-07 | 3.99 | 9.81e-08 | 3.98 | 2.27e-10 | 4.99 | |
| 4 | 34 | 475 | 2.57e-04 | * | 2.65e-03 | * | 1.10e-03 | * | 2.30e-05 | * |
| 134 | 1868 | 8.65e-06 | 4.95 | 6.04e-05 | 5.51 | 3.31e-05 | 5.12 | 2.65e-07 | 6.51 | |
| 502 | 7053 | 3.97e-07 | 4.66 | 2.01e-06 | 5.15 | 9.66e-07 | 5.35 | 4.07e-09 | 6.32 | |
| 1972 | 28028 | 1.18e-08 | 5.14 | 9.16e-08 | 4.52 | 3.73e-08 | 4.76 | 9.82e-11 | 5.44 | |
| 7976 | 113919 | 3.68e-10 | 4.96 | 2.79e-09 | 5.00 | 1.14e-09 | 4.99 | 2.20e-12 | 5.44 |
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