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: October 31, 2019

Revised: January 31, 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.

Keywords:

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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References

[1]	D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.
[2]	F. Brezzi, J. Douglas Jr. and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235. doi: 10.1007/BF01389710.
[3]	B. Li and X. Xie, A two-level algorithm for the weak Galerkin discretization of diffusion problems, J. Comput. Appl. Math., 287 (2015), 179-195. doi: 10.1016/j.cam.2015.03.043.
[4]	K. Liu, L. Song and S. Zhou, An over-penalized weak Galerkin method for second-order elliptic problems, J. Comput. Math., 36 (2018), 866-880. doi: 10.4208/jcm.1705-m2016-0744.
[5]	L. Mu, J. Wang, Y. Wang and X. Ye, A computational study of the weak Galerkin method for second-order elliptic equations, Numer. Algor., 63 (2012), 753-777. doi: 10.1007/s11075-012-9651-1.
[6]	L. Mu, J. Wang, G. Wei, X. Ye and S. Zhao, Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250 (2013), 106-125. doi: 10.1016/j.jcp.2013.04.042.
[7]	L. Mu, J. Wang and X. Ye, A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal., 35 (2015), 1228-1255. doi: 10.1093/imanum/dru026.
[8]	L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math., 285 (2015), 45-58. doi: 10.1016/j.cam.2015.02.001.
[9]	L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31-53.
[10]	L. Mu, J. Wang, X. Ye and S. Zhang, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386. doi: 10.1007/s10915-014-9964-4.
[11]	A. Quarteroni and V. Alberto, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23, Springer, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-540-85268-1.
[12]	P.-A. Raviart and J.-M. Thomas, A mixed finite element method for 2-nd order elliptic problems, in Galligani I., Magenes E. (eds) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol 606, Springer, Berlin, Heidelberg. doi: https://doi.org/10.1007/BFb0064470.
[13]	B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008. doi: doi.org/10.1137/1.9780898717440.
[14]	L. Song, K. Liu and S. Zhao, A weak Galerkin method with an over-relaxed stabilization for low regularity elliptic problems, J. Sci. Comput., 71 (2017), 195-218. doi: 10.1007/s10915-016-0296-4.
[15]	L. Song, S. Zhao and K. Liu, A relaxed weak Galerkin method for elliptic interface problems with low regularity, Appl. Numer. Math., 128 (2018), 65-80. doi: 10.1016/j.apnum.2018.01.021.
[16]	C. Wang and J. Wang, An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68 (2014), 2314-2330. doi: 10.1016/j.camwa.2014.03.021.
[17]	J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115. doi: 10.1016/j.cam.2012.10.003.
[18]	J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126. doi: 10.1090/S0025-5718-2014-02852-4.
[19]	J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174. doi: 10.1007/s10444-016-9471-2.
[20]	Q. Zhai, X. Ye, R. Wang and R. Zhang, A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems, Comput. Math. Appl., 74 (2017), 2243-2252. doi: 10.1016/j.camwa.2017.07.009.