History of convergence of the HDG method for |$\delta =h^{7/4}$| 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 | 36 | 206 | 6.00e-01 | * | 3.19e-01 | * | 9.55e-01 | * | 1.30e+00 | * |
| 140 | 846 | 1.55e-01 | 1.99 | 1.30e-01 | 1.33 | 4.20e-01 | 1.21 | 4.28e-01 | 1.63 | |
| 516 | 3158 | 2.67e-02 | 2.70 | 3.08e-02 | 2.20 | 4.70e-02 | 3.36 | 4.52e-02 | 3.45 | |
| 2014 | 12420 | 4.58e-03 | 2.59 | 6.38e-03 | 2.32 | 7.93e-03 | 2.62 | 7.13e-03 | 2.71 | |
| 8030 | 49628 | 1.11e-03 | 2.05 | 1.36e-03 | 2.23 | 1.17e-03 | 2.77 | 8.67e-04 | 3.05 | |
| 2 | 36 | 295 | 9.86e-02 | * | 1.70e-01 | * | 8.19e-01 | * | 7.83e-01 | * |
| 140 | 1213 | 1.76e-02 | 2.54 | 3.30e-02 | 2.42 | 3.79e-02 | 4.53 | 2.14e-02 | 5.30 | |
| 516 | 4531 | 3.08e-03 | 2.67 | 2.47e-03 | 3.97 | 5.88e-03 | 2.86 | 3.84e-03 | 2.63 | |
| 2014 | 17824 | 1.24e-04 | 4.72 | 1.66e-04 | 3.97 | 1.47e-04 | 5.42 | 5.70e-05 | 6.18 | |
| 8030 | 71233 | 1.05e-05 | 3.57 | 1.66e-05 | 3.33 | 1.09e-05 | 3.76 | 2.11e-06 | 4.77 | |
| 3 | 36 | 384 | 3.61e-02 | * | 3.04e-02 | * | 2.41e-01 | * | 1.55e-01 | * |
| 140 | 1580 | 5.56e-03 | 2.76 | 6.87e-03 | 2.19 | 1.95e-02 | 3.70 | 9.15e-03 | 4.17 | |
| 516 | 5904 | 3.46e-04 | 4.26 | 3.65e-04 | 4.50 | 2.84e-04 | 6.48 | 9.11e-05 | 7.07 | |
| 2014 | 23228 | 4.67e-06 | 6.32 | 1.37e-05 | 4.82 | 1.34e-05 | 4.48 | 6.12e-06 | 3.97 | |
| 8030 | 92838 | 8.97e-08 | 5.71 | 3.65e-07 | 5.25 | 1.62e-07 | 6.39 | 5.98e-08 | 6.69 | |
| 4 | 36 | 473 | 9.84e-02 | * | 1.51e-02 | * | 3.78e-01 | * | 1.51e-01 | * |
| 140 | 1947 | 3.82e-04 | 8.17 | 1.38e-03 | 3.53 | 9.98e-04 | 8.74 | 2.75e-04 | 9.29 | |
| 516 | 7277 | 3.68e-05 | 3.59 | 2.61e-05 | 6.08 | 6.82e-05 | 4.11 | 2.15e-05 | 3.91 | |
| 2014 | 28632 | 1.92e-07 | 7.72 | 2.67e-07 | 6.73 | 3.43e-07 | 7.77 | 1.02e-07 | 7.85 | |
| 8030 | 114443 | 1.63e-09 | 6.90 | 3.04e-09 | 6.48 | 2.79e-09 | 6.96 | 7.23e-10 | 7.16 |
| |$k$| . | |$N$| . | #d.o.f. . | |$e(\mathrm{L})$| . | |$r$| . | |$e(\mathbf{u})$| . | |$r$| . | |$e(p)$| . | |$r$| . | |$\widehat{e}$| . | |$\widehat{r}$| . |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 36 | 206 | 6.00e-01 | * | 3.19e-01 | * | 9.55e-01 | * | 1.30e+00 | * |
| 140 | 846 | 1.55e-01 | 1.99 | 1.30e-01 | 1.33 | 4.20e-01 | 1.21 | 4.28e-01 | 1.63 | |
| 516 | 3158 | 2.67e-02 | 2.70 | 3.08e-02 | 2.20 | 4.70e-02 | 3.36 | 4.52e-02 | 3.45 | |
| 2014 | 12420 | 4.58e-03 | 2.59 | 6.38e-03 | 2.32 | 7.93e-03 | 2.62 | 7.13e-03 | 2.71 | |
| 8030 | 49628 | 1.11e-03 | 2.05 | 1.36e-03 | 2.23 | 1.17e-03 | 2.77 | 8.67e-04 | 3.05 | |
| 2 | 36 | 295 | 9.86e-02 | * | 1.70e-01 | * | 8.19e-01 | * | 7.83e-01 | * |
| 140 | 1213 | 1.76e-02 | 2.54 | 3.30e-02 | 2.42 | 3.79e-02 | 4.53 | 2.14e-02 | 5.30 | |
| 516 | 4531 | 3.08e-03 | 2.67 | 2.47e-03 | 3.97 | 5.88e-03 | 2.86 | 3.84e-03 | 2.63 | |
| 2014 | 17824 | 1.24e-04 | 4.72 | 1.66e-04 | 3.97 | 1.47e-04 | 5.42 | 5.70e-05 | 6.18 | |
| 8030 | 71233 | 1.05e-05 | 3.57 | 1.66e-05 | 3.33 | 1.09e-05 | 3.76 | 2.11e-06 | 4.77 | |
| 3 | 36 | 384 | 3.61e-02 | * | 3.04e-02 | * | 2.41e-01 | * | 1.55e-01 | * |
| 140 | 1580 | 5.56e-03 | 2.76 | 6.87e-03 | 2.19 | 1.95e-02 | 3.70 | 9.15e-03 | 4.17 | |
| 516 | 5904 | 3.46e-04 | 4.26 | 3.65e-04 | 4.50 | 2.84e-04 | 6.48 | 9.11e-05 | 7.07 | |
| 2014 | 23228 | 4.67e-06 | 6.32 | 1.37e-05 | 4.82 | 1.34e-05 | 4.48 | 6.12e-06 | 3.97 | |
| 8030 | 92838 | 8.97e-08 | 5.71 | 3.65e-07 | 5.25 | 1.62e-07 | 6.39 | 5.98e-08 | 6.69 | |
| 4 | 36 | 473 | 9.84e-02 | * | 1.51e-02 | * | 3.78e-01 | * | 1.51e-01 | * |
| 140 | 1947 | 3.82e-04 | 8.17 | 1.38e-03 | 3.53 | 9.98e-04 | 8.74 | 2.75e-04 | 9.29 | |
| 516 | 7277 | 3.68e-05 | 3.59 | 2.61e-05 | 6.08 | 6.82e-05 | 4.11 | 2.15e-05 | 3.91 | |
| 2014 | 28632 | 1.92e-07 | 7.72 | 2.67e-07 | 6.73 | 3.43e-07 | 7.77 | 1.02e-07 | 7.85 | |
| 8030 | 114443 | 1.63e-09 | 6.90 | 3.04e-09 | 6.48 | 2.79e-09 | 6.96 | 7.23e-10 | 7.16 |
History of convergence of the HDG method for |$\delta =h^{7/4}$| 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 | 36 | 206 | 6.00e-01 | * | 3.19e-01 | * | 9.55e-01 | * | 1.30e+00 | * |
| 140 | 846 | 1.55e-01 | 1.99 | 1.30e-01 | 1.33 | 4.20e-01 | 1.21 | 4.28e-01 | 1.63 | |
| 516 | 3158 | 2.67e-02 | 2.70 | 3.08e-02 | 2.20 | 4.70e-02 | 3.36 | 4.52e-02 | 3.45 | |
| 2014 | 12420 | 4.58e-03 | 2.59 | 6.38e-03 | 2.32 | 7.93e-03 | 2.62 | 7.13e-03 | 2.71 | |
| 8030 | 49628 | 1.11e-03 | 2.05 | 1.36e-03 | 2.23 | 1.17e-03 | 2.77 | 8.67e-04 | 3.05 | |
| 2 | 36 | 295 | 9.86e-02 | * | 1.70e-01 | * | 8.19e-01 | * | 7.83e-01 | * |
| 140 | 1213 | 1.76e-02 | 2.54 | 3.30e-02 | 2.42 | 3.79e-02 | 4.53 | 2.14e-02 | 5.30 | |
| 516 | 4531 | 3.08e-03 | 2.67 | 2.47e-03 | 3.97 | 5.88e-03 | 2.86 | 3.84e-03 | 2.63 | |
| 2014 | 17824 | 1.24e-04 | 4.72 | 1.66e-04 | 3.97 | 1.47e-04 | 5.42 | 5.70e-05 | 6.18 | |
| 8030 | 71233 | 1.05e-05 | 3.57 | 1.66e-05 | 3.33 | 1.09e-05 | 3.76 | 2.11e-06 | 4.77 | |
| 3 | 36 | 384 | 3.61e-02 | * | 3.04e-02 | * | 2.41e-01 | * | 1.55e-01 | * |
| 140 | 1580 | 5.56e-03 | 2.76 | 6.87e-03 | 2.19 | 1.95e-02 | 3.70 | 9.15e-03 | 4.17 | |
| 516 | 5904 | 3.46e-04 | 4.26 | 3.65e-04 | 4.50 | 2.84e-04 | 6.48 | 9.11e-05 | 7.07 | |
| 2014 | 23228 | 4.67e-06 | 6.32 | 1.37e-05 | 4.82 | 1.34e-05 | 4.48 | 6.12e-06 | 3.97 | |
| 8030 | 92838 | 8.97e-08 | 5.71 | 3.65e-07 | 5.25 | 1.62e-07 | 6.39 | 5.98e-08 | 6.69 | |
| 4 | 36 | 473 | 9.84e-02 | * | 1.51e-02 | * | 3.78e-01 | * | 1.51e-01 | * |
| 140 | 1947 | 3.82e-04 | 8.17 | 1.38e-03 | 3.53 | 9.98e-04 | 8.74 | 2.75e-04 | 9.29 | |
| 516 | 7277 | 3.68e-05 | 3.59 | 2.61e-05 | 6.08 | 6.82e-05 | 4.11 | 2.15e-05 | 3.91 | |
| 2014 | 28632 | 1.92e-07 | 7.72 | 2.67e-07 | 6.73 | 3.43e-07 | 7.77 | 1.02e-07 | 7.85 | |
| 8030 | 114443 | 1.63e-09 | 6.90 | 3.04e-09 | 6.48 | 2.79e-09 | 6.96 | 7.23e-10 | 7.16 |
| |$k$| . | |$N$| . | #d.o.f. . | |$e(\mathrm{L})$| . | |$r$| . | |$e(\mathbf{u})$| . | |$r$| . | |$e(p)$| . | |$r$| . | |$\widehat{e}$| . | |$\widehat{r}$| . |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 36 | 206 | 6.00e-01 | * | 3.19e-01 | * | 9.55e-01 | * | 1.30e+00 | * |
| 140 | 846 | 1.55e-01 | 1.99 | 1.30e-01 | 1.33 | 4.20e-01 | 1.21 | 4.28e-01 | 1.63 | |
| 516 | 3158 | 2.67e-02 | 2.70 | 3.08e-02 | 2.20 | 4.70e-02 | 3.36 | 4.52e-02 | 3.45 | |
| 2014 | 12420 | 4.58e-03 | 2.59 | 6.38e-03 | 2.32 | 7.93e-03 | 2.62 | 7.13e-03 | 2.71 | |
| 8030 | 49628 | 1.11e-03 | 2.05 | 1.36e-03 | 2.23 | 1.17e-03 | 2.77 | 8.67e-04 | 3.05 | |
| 2 | 36 | 295 | 9.86e-02 | * | 1.70e-01 | * | 8.19e-01 | * | 7.83e-01 | * |
| 140 | 1213 | 1.76e-02 | 2.54 | 3.30e-02 | 2.42 | 3.79e-02 | 4.53 | 2.14e-02 | 5.30 | |
| 516 | 4531 | 3.08e-03 | 2.67 | 2.47e-03 | 3.97 | 5.88e-03 | 2.86 | 3.84e-03 | 2.63 | |
| 2014 | 17824 | 1.24e-04 | 4.72 | 1.66e-04 | 3.97 | 1.47e-04 | 5.42 | 5.70e-05 | 6.18 | |
| 8030 | 71233 | 1.05e-05 | 3.57 | 1.66e-05 | 3.33 | 1.09e-05 | 3.76 | 2.11e-06 | 4.77 | |
| 3 | 36 | 384 | 3.61e-02 | * | 3.04e-02 | * | 2.41e-01 | * | 1.55e-01 | * |
| 140 | 1580 | 5.56e-03 | 2.76 | 6.87e-03 | 2.19 | 1.95e-02 | 3.70 | 9.15e-03 | 4.17 | |
| 516 | 5904 | 3.46e-04 | 4.26 | 3.65e-04 | 4.50 | 2.84e-04 | 6.48 | 9.11e-05 | 7.07 | |
| 2014 | 23228 | 4.67e-06 | 6.32 | 1.37e-05 | 4.82 | 1.34e-05 | 4.48 | 6.12e-06 | 3.97 | |
| 8030 | 92838 | 8.97e-08 | 5.71 | 3.65e-07 | 5.25 | 1.62e-07 | 6.39 | 5.98e-08 | 6.69 | |
| 4 | 36 | 473 | 9.84e-02 | * | 1.51e-02 | * | 3.78e-01 | * | 1.51e-01 | * |
| 140 | 1947 | 3.82e-04 | 8.17 | 1.38e-03 | 3.53 | 9.98e-04 | 8.74 | 2.75e-04 | 9.29 | |
| 516 | 7277 | 3.68e-05 | 3.59 | 2.61e-05 | 6.08 | 6.82e-05 | 4.11 | 2.15e-05 | 3.91 | |
| 2014 | 28632 | 1.92e-07 | 7.72 | 2.67e-07 | 6.73 | 3.43e-07 | 7.77 | 1.02e-07 | 7.85 | |
| 8030 | 114443 | 1.63e-09 | 6.90 | 3.04e-09 | 6.48 | 2.79e-09 | 6.96 | 7.23e-10 | 7.16 |
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