Theory and Modern Applications
Table 1 Error estimates and convergence rates
h , Δ t | \(\boldsymbol{\|u-u_{h}\|_{L^{\infty}(L^{2}(\Omega))}}\) | Rate | \(\boldsymbol{\| \boldsymbol{\sigma}-\boldsymbol{\sigma}_{h}\|_{L^{\infty}((L^{2}(\Omega ))^{2})}}\) | Rate |
|---|---|---|---|---|
\((\frac{\sqrt{2}}{8},\frac{1}{8})\) | 3.8464e − 003 | 1.6043e − 002 | ||
\((\frac{\sqrt{2}}{16},\frac{1}{16})\) | 2.0591e − 003 | 0.90 | 8.6877e − 003 | 0.88 |
\((\frac{\sqrt{2}}{32},\frac{1}{32})\) | 1.0637e − 003 | 0.95 | 4.5181e − 003 | 0.94 |
\((\frac{\sqrt{2}}{64},\frac{1}{64})\) | 5.4043e − 004 | 0.98 | 2.3037e − 003 | 0.97 |
h , Δ t | \(\boldsymbol{\|\lambda-\lambda_{h}\| _{L^{\infty}((L^{2}(\Omega))^{2})}}\) | Rate | \(\boldsymbol{\|\lambda -\lambda\|_{L^{\infty}(H(\operatorname{div},\Omega))}}\) | Rate |
|---|---|---|---|---|
\((\frac{\sqrt{2}}{8},\frac{1}{8})\) | 1.5432e − 002 | 5.1929e − 002 | ||
\((\frac{\sqrt{2}}{16},\frac{1}{16})\) | 8.3902e − 003 | 0.88 | 2.7873e − 002 | 0.90 |
\((\frac{\sqrt{2}}{32},\frac{1}{32})\) | 4.3491e − 003 | 0.95 | 1.4411e − 002 | 0.95 |
\((\frac{\sqrt{2}}{64},\frac{1}{64})\) | 2.2108e − 003 | 0.98 | 7.3231e − 003 | 0.98 |