Article Contents

2024, Volume 29, Issue 4: 1652-1669. Doi: 10.3934/dcdsb.2023149

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A new over-penalized weak Galerkin method. Part Ⅲ: Convection-diffusion-reaction problems

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

^* Corresponding author: Lunji Song
^* Corresponding author: Lunji Song

Received: May 2022

Revised: August 2023

Early access: September 2023

Published: April 2024

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Abstract

In this paper, we propose an over-penalized weak Galerkin (OPWG) finite element method for stationary convection-diffusion-reaction equations with full variable coefficients. This method employs piecewise polynomial approximations of degree $ k $ ($ k\geq 1 $) for both the scalar function and its trace. Especially, the trace on inter-element boundaries is approximated by double-valued functions instead of single-valued ones. The $ (\mathbb{P}_{k}, \mathbb{P}_{k},[\mathbb{P}_{k-1}]^{d}) $ elements, with dimensions of space $ d = 2,\; 3 $ are employed. Our method deals with the convective term discretized in a trilinear form, and the uniqueness of numerical solutions is discussed. Optimal error estimates in the discrete $ H^1 $-norm and $ L^2 $-norm are established, from which the optimal penalty exponent can be fixed. Numerical examples confirm the theory.

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

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Figure 1. The initial meshes: (a) unit square; (b) L-shaped domain

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Figure 2. Example 3. The plot of WG solution (Left) for $k = 2, \beta_0 = 5$ and $h = 1/128$ and the exact solution (Right)

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Table 1. Example 1. Numerical errors and convergence rates

$ k=1 $	$ h $	$ \|\|\|e_h\|\|\| $	rates	$ \|\|e_0\|\| $	rates
$ \beta_0=2 $	1/4	1.3064e-01	—	7.0262e-01	—
	1/8	9.1251e-02	0.5176	3.3170e-01	1.0828
	1/16	6.4970e-02	0.4900	1.6448e-01	1.0119
	1/32	4.6338e-02	0.4875	8.2620e-02	0.9933
	1/64	3.2963e-02	0.4913	4.1529e-02	0.9923
	1/128	2.3390e-02	0.4949	2.0838e-02	0.9949
$ \beta_0=3 $	1/4	1.0541e-01	—	4.5887e-01	—
	1/8	5.5024e-02	0.9378	1.2313e-01	1.8979
	1/16	2.8019e-02	0.9736	3.1732e-02	1.9561
	1/32	1.4121e-02	0.9885	8.0389e-03	1.9808
	1/64	7.0857e-03	0.9948	2.0220e-03	1.9912
	1/128	3.5489e-03	0.9975	5.0698e-04	1.9957
$ k=2 $	$ h $	$ \|\|\|e_h\|\|\| $	rates	$ \|\|e_0\|\| $	rates
$ \beta_0=4 $	1/4	3.8617e-02	—	6.4250e-02	—
	1/8	1.4377e-02	1.4254	8.5362e-03	2.9120
	1/16	5.1701e-03	1.4754	1.0891e-03	2.9704
	1/32	1.8422e-03	1.4887	1.3752e-04	2.9854
	1/64	6.5388e-04	1.4943	1.7279e-05	2.9925
	1/128	2.3164e-04	1.4971	2.0897e-06	3.0476
$ \beta_0=5 $	1/4	2.4510e-02	—	3.3695e-02	—
	1/8	6.3590e-03	1.9464	3.4220e-03	3.2996
	1/16	1.6140e-03	1.9781	3.9543e-04	3.1133
	1/32	4.0656e-04	1.9891	4.8273e-05	3.0341
	1/64	1.0203e-04	1.9944	6.0361e-06	2.9995
	1/128	2.7533e-05	1.8897	2.3620e-05	-1.9683

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Table 2. Example 2. L-shaped domain

	$ h $	$ \|\|\|e_h\|\|\| $	rates	$ \|\|e_0\|\| $	rates
$ k=1 $ $ \beta_0=3 $	1/4	1.1333e+01	—	3.7667e+00	—
	1/8	6.1262e+00	0.8874	1.0837e+00	1.7973
	1/16	3.1637e+00	0.9533	2.8710e-01	1.9163
	1/32	1.6016e+00	0.9820	7.3398e-02	1.9677
	1/64	8.0463e-01	0.9931	1.8508e-02	1.9875
	1/128	4.0310e-01	0.9971	4.6435e-03	1.9948
$ k=2 $ $ \beta_0=5 $	1/4	5.4298e+00	—	9.7421e-01	—
	1/8	1.5931e+00	1.7973	1.0745e-01	3.1805
	1/16	4.0802e-01	1.9163	1.2024e-02	3.1596
	1/32	1.0273e-01	1.9677	1.4513e-03	3.0504
	1/64	2.5764e-02	1.9875	1.7971e-04	3.0136
	1/128	6.4507e-03	1.9948	2.2406e-05	3.0037

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Table 3. Example 3. Interior layer

	$ h $	$ \|\|\|e_h\|\|\| $	rates	$ \|\|e_0\|\| $	rates
$ k=1 $ $ \beta_0=3 $	1/4	1.9364e-02	—	1.3731e-01	—
	1/8	1.0828e-02	0.8386	7.2267e-02	0.9260
	1/16	5.3065e-03	1.0289	1.3719e-02	2.3971
	1/32	2.3762e-03	1.1591	2.5532e-03	2.4257
	1/64	1.1754e-03	1.0155	6.3756e-04	2.0016
$ k=2 $ $ \beta_0=5 $	1/4	1.0119e-02	—	7.7888e-02	—
	1/8	2.8314e-03	1.8374	9.7203e-03	3.0023
	1/16	7.6948e-04	1.8795	9.5727e-04	3.3440
	1/32	2.5220e-04	1.6093	1.2603e-04	2.9251
	1/64	6.6223e-05	1.9291	1.7346e-05	2.8610

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