American Institute of Mathematical Sciences

May  2021, 26(5): 2411-2428. doi: 10.3934/dcdsb.2020184

A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems

 1 School of Mathematics and Statistics, and Key Laboratory of Applied Mathematics and Complex Systems (Gansu Province), Lanzhou University, Lanzhou 730000, China 2 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA

* Corresponding author: Lunji Song

Received  November 2019 Revised  February 2020 Published  May 2021 Early access  June 2020

In this article, we propose a new over-penalized weak Galerkin (OPWG) method with a stabilizer for second-order elliptic problems. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements $(\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{d})$, or $(\mathbb{P}_{k}, \mathbb{P}_{k-1}, [\mathbb{P}_{k-1}]^{d})$, with dimensions of space $d = 2, \; 3$. The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shape-regularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are $O(h^{-\beta_{0}(d-1)-1})$, $\beta_{0}$ being the penalty exponent. Therefore we introduce a new mini-block diagonal preconditioner, which is proven to be theoretically and numerically effective in reducing the condition numbers of stiffness matrices to the magnitude of $O(h^{-2})$. Optimal error estimates in a discrete $H^1$-norm and $L^2$-norm are established, from which the optimal penalty exponent can be easily chosen. Several numerical examples are presented to demonstrate flexibility, effectiveness and reliability of the new method.

Citation: Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
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References:
Initial mesh (Left) and the OPWG ($\beta_{0} = 3$) solution on the finest mesh (Right) for Example 2
Convergence rates of the OPWG ($\beta_{0} = 3$) solutions against degree of freedoms for different values of $\alpha$ in Example 2. (Left) $\alpha = 0.5$; (Right) $\alpha = 0.25$
WG method with element $(\mathbb{P}_k, \mathbb{P}_{k}, \mathbb{P}_{k-1}^2)$ for Example 1
 $h$ $k=1$ $k=2$ $||| e_{h}|||$ $\|e_{0}\|$ $||| e_{h}|||$ $\|e_{0}\|$ 1/8 9.9173e-01 5.3131e-02 7.8508e-02 3.2464e-03 1/16 4.9588e-01 1.3272e-02 1.9677e-02 4.0611e-04 1/32 2.4793e-01 3.3169e-03 4.9226e-03 5.0761e-05 1/64 1.2396e-01 8.2914e-04 1.2309e-03 6.3445e-06 Rate. 1.0001 2.0001 1.9997 3.0001
 $h$ $k=1$ $k=2$ $||| e_{h}|||$ $\|e_{0}\|$ $||| e_{h}|||$ $\|e_{0}\|$ 1/8 9.9173e-01 5.3131e-02 7.8508e-02 3.2464e-03 1/16 4.9588e-01 1.3272e-02 1.9677e-02 4.0611e-04 1/32 2.4793e-01 3.3169e-03 4.9226e-03 5.0761e-05 1/64 1.2396e-01 8.2914e-04 1.2309e-03 6.3445e-06 Rate. 1.0001 2.0001 1.9997 3.0001
OPWG with $(\mathbb{P}_1, \mathbb{P}_1, \mathbb{P}_{0}^{2})$ and $\beta_0 = 1, 2, 3, 4$ for Example 1
 $h$ $||| e_h |||$ $\|e_0\|$ $||| e_h |||$ $\|e_0\|$ $\beta_0=1$ $\beta_0=2$ $1/8$ 1.3258e+00 7.4931e-02 1.0550e+00 5.9839e-02 $1/16$ 1.0306e+00 3.4203e-02 5.6097e-01 1.5700e-02 $1/32$ 9.4663e-01 2.5074e-02 3.1098e-01 4.3112e-03 $1/64$ 9.2663e-01 2.3053e-02 1.8215e-01 1.2815e-03 Rate. 0.0308 0.1212 0.7717 1.7503 $\beta_0=3$ $\beta_0=4$ $1/8$ 1.0019e+00 5.7508e-02 9.9304e-01 5.7135e-02 $1/16$ 5.0137e-01 1.4381e-02 4.9628e-01 1.4278e-02 $1/32$ 2.5075e-01 3.5954e-03 2.4804e-01 3.5682e-03 $1/64$ 1.2539e-01 8.9887e-04 1.2399e-01 8.9191e-04 Rate. 0.9998 2. 1.0003 2.0002
 $h$ $||| e_h |||$ $\|e_0\|$ $||| e_h |||$ $\|e_0\|$ $\beta_0=1$ $\beta_0=2$ $1/8$ 1.3258e+00 7.4931e-02 1.0550e+00 5.9839e-02 $1/16$ 1.0306e+00 3.4203e-02 5.6097e-01 1.5700e-02 $1/32$ 9.4663e-01 2.5074e-02 3.1098e-01 4.3112e-03 $1/64$ 9.2663e-01 2.3053e-02 1.8215e-01 1.2815e-03 Rate. 0.0308 0.1212 0.7717 1.7503 $\beta_0=3$ $\beta_0=4$ $1/8$ 1.0019e+00 5.7508e-02 9.9304e-01 5.7135e-02 $1/16$ 5.0137e-01 1.4381e-02 4.9628e-01 1.4278e-02 $1/32$ 2.5075e-01 3.5954e-03 2.4804e-01 3.5682e-03 $1/64$ 1.2539e-01 8.9887e-04 1.2399e-01 8.9191e-04 Rate. 0.9998 2. 1.0003 2.0002
OPWG with $(\mathbb{P}_2, \mathbb{P}_2, \mathbb{P}_{1}^{2})$ and $\beta_0 = 2, 3, 4, 5$ for Example 1
 $h$ $||| e_h |||$ $\|e_0\|$ $||| e_h |||$ $\|e_0\|$ $\beta_0=2$ $\beta_0=3$ $1/8$ 1.7917e+00 1.0256e-01 8.4595e-01 2.0487e-02 $1/16$ 1.4060e+00 5.7784e-02 4.4264e-01 5.3334e-03 $1/32$ 1.0580e+00 3.0997e-02 2.2464e-01 1.3530e-03 $1/64$ 7.7468e-01 1.6118e-02 1.1291e-01 3.4023e-04 Rate. 0.4497 0.9435 0.9924 1.9916 $\beta_0=4$ $\beta_0=5$ $1/8$ 3.6348e-01 4.6229e-03 1.6594e-01 3.2492e-03 $1/16$ 1.2933e-01 5.9131e-04 4.1839e-02 4.0604e-04 $1/32$ 4.5730e-02 7.4938e-05 1.0494e-02 5.0757e-05 $1/64$ 1.6162e-02 9.4394e-06 2.6274e-03 6.3443e-06 Rate. 1.5005 2.9889 1.9979 3.0001
 $h$ $||| e_h |||$ $\|e_0\|$ $||| e_h |||$ $\|e_0\|$ $\beta_0=2$ $\beta_0=3$ $1/8$ 1.7917e+00 1.0256e-01 8.4595e-01 2.0487e-02 $1/16$ 1.4060e+00 5.7784e-02 4.4264e-01 5.3334e-03 $1/32$ 1.0580e+00 3.0997e-02 2.2464e-01 1.3530e-03 $1/64$ 7.7468e-01 1.6118e-02 1.1291e-01 3.4023e-04 Rate. 0.4497 0.9435 0.9924 1.9916 $\beta_0=4$ $\beta_0=5$ $1/8$ 3.6348e-01 4.6229e-03 1.6594e-01 3.2492e-03 $1/16$ 1.2933e-01 5.9131e-04 4.1839e-02 4.0604e-04 $1/32$ 4.5730e-02 7.4938e-05 1.0494e-02 5.0757e-05 $1/64$ 1.6162e-02 9.4394e-06 2.6274e-03 6.3443e-06 Rate. 1.5005 2.9889 1.9979 3.0001
Comparison of condition number with optimal penalty parameters
 $h$ Without preconditioning Block-diagonal preconditioning $k=1, \beta_{0}=3$ $k=2, \beta_{0}=5$ $k=1, \beta_{0}=3$ $k=2, \beta_{0}=5$ 1/4 1.8272e+04 2.1075e+04 2.6265e+02 4.3079e+04 1/8 1.6957e+05 6.9533e+05 1.0135e+03 1.9257e+05 1/16 2.1585e+06 3.4759e+07 4.0481e+03 8.0866e+05 1/32 3.2471e+07 2.0653e+09 1.6228e+04 3.3427e+06 1/64 5.1218e+08 1.2974e+11 6.4995e+04 1.3570e+07 Order -3.9794 -5.9731 -2.0018 -2.0213
 $h$ Without preconditioning Block-diagonal preconditioning $k=1, \beta_{0}=3$ $k=2, \beta_{0}=5$ $k=1, \beta_{0}=3$ $k=2, \beta_{0}=5$ 1/4 1.8272e+04 2.1075e+04 2.6265e+02 4.3079e+04 1/8 1.6957e+05 6.9533e+05 1.0135e+03 1.9257e+05 1/16 2.1585e+06 3.4759e+07 4.0481e+03 8.0866e+05 1/32 3.2471e+07 2.0653e+09 1.6228e+04 3.3427e+06 1/64 5.1218e+08 1.2974e+11 6.4995e+04 1.3570e+07 Order -3.9794 -5.9731 -2.0018 -2.0213
Errors and condition numbers for Example 1 with $(\mathbb{P}_1, \mathbb{P}_0, \mathbb{P}_{0}^{2})$ and $\beta_0 = 3$
 $h$ $||| e_h |||$ $\|e_0\|$ Cond. Pre. Cond. $1/8$ 1.5226e+00 8.7459e-02 6.6551e+03 2.1055e+03 $1/16$ 7.7405e-01 2.2035e-02 8.1739e+04 7.9132e+03 $1/32$ 3.8921e-01 5.5253e-03 1.2109e+06 3.1176e+04 $1/64$ 1.9499e-01 1.3832e-03 1.9007e+07 1.2425e+05 Rate. 0.9971 1.9980 -3.9724 -1.9947
 $h$ $||| e_h |||$ $\|e_0\|$ Cond. Pre. Cond. $1/8$ 1.5226e+00 8.7459e-02 6.6551e+03 2.1055e+03 $1/16$ 7.7405e-01 2.2035e-02 8.1739e+04 7.9132e+03 $1/32$ 3.8921e-01 5.5253e-03 1.2109e+06 3.1176e+04 $1/64$ 1.9499e-01 1.3832e-03 1.9007e+07 1.2425e+05 Rate. 0.9971 1.9980 -3.9724 -1.9947
Errors and condition numbers for Example 1 with $(\mathbb{P}_2, \mathbb{P}_1, \mathbb{P}_{1}^{2})$ and $\beta_0 = 5$
 $h$ $||| e_h |||$ $\|e_0\|$ Cond. Pre. Cond. $1/8$ 1.6622e-01 3.3354e-03 3.5419e+05 1.0229e+05 $1/16$ 4.1912e-02 4.1731e-04 1.9810e+07 4.2176e+05 $1/32$ 1.0513e-02 5.2184e-05 1.2227e+09 1.7166e+06 $1/64$ 2.6321e-03 6.5231e-06 7.7586e+10 6.9742e+06 Rate. 1.9979 3.0000 -5.9877 -2.0225
 $h$ $||| e_h |||$ $\|e_0\|$ Cond. Pre. Cond. $1/8$ 1.6622e-01 3.3354e-03 3.5419e+05 1.0229e+05 $1/16$ 4.1912e-02 4.1731e-04 1.9810e+07 4.2176e+05 $1/32$ 1.0513e-02 5.2184e-05 1.2227e+09 1.7166e+06 $1/64$ 2.6321e-03 6.5231e-06 7.7586e+10 6.9742e+06 Rate. 1.9979 3.0000 -5.9877 -2.0225
OPWG with $(\mathbb{P}_1, \mathbb{P}_1, \mathbb{P}_{0}^{2})$ and optimal penalty parameter for Example 2
 dof. $\alpha=0.5$ $\alpha=0.25$ Condition Number $||| e_h |||$ $\|e_{0}\|$ $||| e_h |||$ $\|e_{0}\|$ Cond. Pre. Cond. 2.0880e+3 4.2303e-1 7.0060e-2 9.6614e-1 7.3906e-2 1.7335e+6 1.1357e+3 8.3520e+3 2.7461e-1 1.8365e-2 7.8292e-1 2.2116e-2 2.4882e+7 4.5578e+3 3.3408e+4 1.8503e-1 4.7471e-3 6.4655e-1 7.3419e-3 3.8905e+8 1.8270e+4 1.3363e+5 1.3025e-1 1.2371e-3 5.5838e-1 2.8274e-3 6.1854e+9 7.3140e+4 5.3453e+5 9.2125e-2 3.2954e-4 4.6994e-1 1.1394e-3 9.8828e+10 2.9267e+5
 dof. $\alpha=0.5$ $\alpha=0.25$ Condition Number $||| e_h |||$ $\|e_{0}\|$ $||| e_h |||$ $\|e_{0}\|$ Cond. Pre. Cond. 2.0880e+3 4.2303e-1 7.0060e-2 9.6614e-1 7.3906e-2 1.7335e+6 1.1357e+3 8.3520e+3 2.7461e-1 1.8365e-2 7.8292e-1 2.2116e-2 2.4882e+7 4.5578e+3 3.3408e+4 1.8503e-1 4.7471e-3 6.4655e-1 7.3419e-3 3.8905e+8 1.8270e+4 1.3363e+5 1.3025e-1 1.2371e-3 5.5838e-1 2.8274e-3 6.1854e+9 7.3140e+4 5.3453e+5 9.2125e-2 3.2954e-4 4.6994e-1 1.1394e-3 9.8828e+10 2.9267e+5
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