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

Kaifang Liu; Lunji Song; Shan Zhao

doi:10.3934/dcdsb.2020184

Article Contents

2021, Volume 26, Issue 5: 2411-2428. Doi: 10.3934/dcdsb.2020184

This issue Previous Article A nonisothermal thermodynamical model of liquid-vapor interaction with metastability Next Article Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations

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
^* Corresponding author: Lunji Song

Received: November 2019

Revised: February 2020

Early access: June 2020

Published: May 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 35j50.

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Figure 1. Initial mesh (Left) and the OPWG ($ \beta_{0} = 3 $) solution on the finest mesh (Right) for Example 2

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Figure 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 $

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Table 1. 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

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Table 2. 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

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Table 3. 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

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Table 4. 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

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Table 5. 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

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Table 6. 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

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Table 7. 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

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