Article Contents
Article Contents

# Weak Galerkin method for the Stokes equations with damping

• * Corresponding author: Qilong Zhai

The research is supported by China Natural National Science Foundation(12001232, 11901015)

• In this paper, we introduce the weak Galerkin (WG) finite element method for the Stokes equations with damping. We establish the WG numerical scheme on general meshes and prove the well-posedness of the scheme. Optimal error estimates for the velocity and pressure are derived. Furthermore, in order to accelerate the WG algorithm, we present a two-level method and give the corresponding error estimates. Finally, some numerical examples are reported to validate the theoretical analysis.

Mathematics Subject Classification: Primary, 65N30, 65N15, 65N12; Secondary, 35B45, 35J50.

 Citation:

• Figure 1.  Left panel: The vectorgraph of velocity; Right panel: The streamlines of velocity

Figure 2.  Left panel: The vectorgraph of velocity; Right panel: The streamlines of velocity

Table 1.  The errors and the order of convergence by one-level method, for $(7.1)$

 h ${|\!|\!|} Q_h \boldsymbol{u}- \boldsymbol{u}_{h} {|\!|\!|}$ order $\|Q_0 {{\mathbf{u}}}- {{\mathbf{u}}}_0\|_0$ order $\|Q_h p-p_{h}\|_0$ order $\frac{1}{16}$ 5.3818$E-$01 - 1.8464$E-$02 - 1.3715$E-$01 - $\frac{1}{25}$ 3.4671$E-$01 0.99 7.6837$E-$03 1.96 6.5631$E-$02 1.65 $\frac{1}{64}$ 1.3613$E-$01 0.99 1.1871$E-$03 1.99 1.2856$E-$02 1.73 $\frac{1}{81}$ 1.0761$E-$01 1.0 7.4195$E-$04 2.0 8.4724$E-$03 1.77

Table 2.  The errors and the order of convergence by two-level method, for $(7.1)$

 (H, h) ${|\!|\!|} Q_h \boldsymbol{u}- \boldsymbol{u}_{h} {|\!|\!|}$ order $\|Q_0 {{\mathbf{u}}}- {{\mathbf{u}}}_0\|_0$ order $\|Q_h p-p_{h}\|_0$ order ($\frac14$, $\frac{1}{16}$) 5.4020$E-$01 - 1.8559$E-$02 - 1.2727$E-$01 - ($\frac15$, $\frac{1}{25})$ 3.4732$E-$01 0.99 7.7109$E-$03 1.97 6.1089$E-$02 1.64 ($\frac{1}{8}$, $\frac{1}{64})$ 1.3618$E-$01 1.0 1.1916$E-$03 1.99 1.2175$E-$02 1.72 ($\frac{1}{9}$, $\frac{1}{81})$ 1.0763$E-$01 1.0 7.4504$E-$04 1.99 8.0646$E-$03 1.75

Table 3.  The comparison of the CPU times (unit:s) for the one-level and two-level methods for $(7.1)$

 h one-level two level ($\frac14$, $\frac{1}{16}$) 6.921 9.352 ($\frac{1}{5}$, $\frac{1}{25})$ 19.923 24.005 ($\frac{1}{8}$, $\frac{1}{64})$ 319.522 277.700 ($\frac{1}{9}$, $\frac{1}{81})$ 974.475 691.726

Table 4.  The comparison of the CPU times (unit:s) for the one-level and two-level methods with different pairs of parameters, for $(7.2)$

 one-level two level $\alpha=1$, $r=3$, $\mu=1$ 308.325 278.230 $\alpha=5$, $r=3$, $\mu=1$ 427.230 276.781 $\alpha=10$, $r=3$, $\mu=1$ 425.516 277.020 $\alpha=1$, $r=4$, $\mu=1$ 314.077 276.544 $\alpha=1$, $r=5$, $\mu=1$ 371.219 277.493 $\alpha=1$, $r=3$, $\mu=0.1$ 476.156 285.622
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