Table 7 Convergence orders of the standard Galerkin FEM at time \(\pmb{T=1.0}\) with time step \(\pmb{\Delta t=0.01}\)
From: The Euler implicit/explicit scheme for the Boussinesq equations
\(\boldsymbol {\frac{1}{h}}\) | \(\boldsymbol {\|u^{n,h}-u^{n,\frac{h}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{u,h,0}}\) | \(\boldsymbol {\|u^{n,h}-u^{n,\frac{h}{2}}\|_{1}}\) | \(\boldsymbol {\rho_{u,h,1}}\) | \(\boldsymbol {\|p^{n,h}-p^{n,\frac{h}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{p,h,0}}\) |
|---|---|---|---|---|---|---|
2 | 0.0172184 | 2.65194 | 0.230412 | 2.14169 | 0.63202 | 4.86123 |
4 | 0.00649278 | 3.37905 | 0.107585 | 2.20926 | 0.130012 | 3.19887 |
8 | 0.00192148 | 3.98286 | 0.0486971 | 2.12864 | 0.0406432 | 3.29174 |
16 | 0.000482436 | 0.0228771 | 0.012347 |
\(\boldsymbol {\frac{1}{h}}\) | \(\boldsymbol {\|\theta^{n,h}-\theta^{n,\frac{h}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{\theta,h,0}}\) | \(\boldsymbol {\|\theta^{n,h}-\theta^{n,\frac{h}{2}}\|_{1}}\) | \(\boldsymbol {\rho_{\theta,h,1}}\) |
|---|---|---|---|---|
2 | 0.0166398 | 3.77383 | 0.131697 | 1.96256 |
4 | 0.00440926 | 3.80314 | 0.0671046 | 1.91516 |
8 | 0.00115937 | 3.92768 | 0.0350386 | 1.99231 |
16 | 0.000295181 | 0.0175869 |