A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

Xiu Ye; Shangyou Zhang

doi:10.3934/dcdsb.2020277

Article Contents

2021, Volume 26, Issue 8: 4131-4145. Doi: 10.3934/dcdsb.2020277

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A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

Xiu Ye^1, and
Shangyou Zhang^2, ,

1.
Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA
2.
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

^* Corresponding author: Shangyou Zhang
^* Corresponding author: Shangyou Zhang

Received: March 31, 2020

Revised: July 31, 2020

Early access: September 2020

Published: August 2021

The first author was supported in part by National Science Foundation Grant DMS-1620016

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Abstract

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $ H^1 $ norm and the standard $ L^2 $ norm. The numerical examples are tested on various meshes and confirm the theory.

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

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Figure 5.1. Example 1. The first three triangular grids

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Figure 5.2. Example 2. The first three triangular grids

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Figure 5.3. Example 3. The first three rectangular grids

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Figure 5.4. Example 4. The first three polygonal grids

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Table 1.1. Weak gradient calculated by (1.4), $ {{|||}}\cdot{{|||}} = O(h^{r_1}) $ and $ \|\cdot\| = O(h^{r_2}) $

$ P_{k}(T) $	$ P_{k-1}(e) $	$ [P_{k+1}(T)]^d $	$ r_1 $	$ r_2 $
$ P_1(T) $	$ P_0(e) $	$ [P_2(T)]^2 $	$ 0 $	$ 0 $
$ P_2(T) $	$ P_1(e) $	$ [P_3(T)]^2 $	$ 1 $	$ 2 $
$ P_3(T) $	$ P_2(e) $	$ [P_4(T)]^2 $	$ 2 $	$ 3 $

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Table 1.2. Weak gradient calculated by (2.3), $ {{|||}}\cdot{{|||}} = O(h^{r_1}) $ and $ \|\cdot\| = O(h^{r_2}) $

$ P_{k}(T) $	$ P_{k-1}(e) $	$ [P_{k+1}(T)]^d $	$ r_1 $	$ r_2 $
$ P_1(T) $	$ P_0(e) $	$ [P_2(T)]^2 $	$ 1 $	$ 2 $
$ P_2(T) $	$ P_1(e) $	$ [P_3(T)]^2 $	$ 2 $	$ 3 $
$ P_3(T) $	$ P_2(e) $	$ [P_4(T)]^2 $	$ 3 $	$ 4 $

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Table 5.1. Example 1. The $ P_k-P_{k-1}-[P_{k+1}]^2 $ element, on triangular grids shown in Figure 5.1

$ k $	$ {{\mathcal T}}_l $	$ {{\|\|\|}} Q_hu-u_h{{\|\|\|}} $	Rate	$ \\|Q_hu-u_h\\| $	Rate
	6	0.3871E-01	1.00	0.3306E-03	1.99
1	7	0.1937E-01	1.00	0.8279E-04	2.00
	8	0.9685E-02	1.00	0.2070E-04	2.00
	6	0.4131E-03	1.98	0.1783E-05	2.95
2	7	0.1038E-03	1.99	0.2268E-06	2.97
	8	0.2602E-04	2.00	0.2859E-07	2.99
	5	0.2925E-04	2.99	0.1515E-06	3.98
3	6	0.3665E-05	3.00	0.9518E-08	3.99
	7	0.4587E-06	3.00	0.5963E-09	4.00
	5	0.4091E-06	3.99	0.1592E-08	4.97
4	6	0.2568E-07	3.99	0.5026E-10	4.99
	7	0.1608E-08	4.00	0.1610E-11	4.96

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Table 5.2. Example 2. The $ P_k-P_{k-1}-[P_{k+1}]^2 $ element, on rectangular grids shown in Figure 5.2

$ k $	$ {{\mathcal T}}_l $	$ {{\|\|\|}} Q_hu-u_h{{\|\|\|}} $	Rate	$ \\|Q_hu-u_h\\| $	Rate
	6	0.120E-02	2.01	0.141E-01	2.00
1	7	0.300E-03	2.00	0.354E-02	2.00
	8	0.749E-04	2.00	0.885E-03	2.00
	6	0.5734E-03	2.00	0.1482E-05	3.00
2	7	0.1434E-03	2.00	0.1852E-06	3.00
	8	0.3584E-04	2.00	0.2314E-07	3.00
	5	0.3645E-04	3.00	0.1360E-06	3.99
3	6	0.4559E-05	3.00	0.8517E-08	4.00
	7	0.5699E-06	3.00	0.5326E-09	4.00
	5	0.4222E-06	4.00	0.9119E-09	5.01
4	6	0.2639E-07	4.00	0.2846E-10	5.00
	7	0.1650E-08	4.00	0.1110E-11	4.68

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Table 5.3. Example 3. The $ P_k-P_{k-1}-[P_{k+1}]^2 $ element, on rectangular grids shown in Figure 5.3

$ k $	$ {{\mathcal T}}_l $	$ {{\|\|\|}} Q_hu-u_h{{\|\|\|}} $	Rate	$ \\|Q_hu-u_h\\| $	Rate
	6	0.141E-01	2.00	0.120E-02	2.01
1	7	0.354E-02	2.00	0.300E-03	2.00
	8	0.885E-03	2.00	0.749E-04	2.00
	6	0.158E-03	3.00	0.113E-05	4.00
2	7	0.197E-04	3.00	0.709E-07	4.00
	8	0.246E-05	3.00	0.444E-08	4.00
	4	0.251E-03	4.80	0.409E-05	5.28
3	5	0.143E-04	4.13	0.128E-06	4.99
	6	0.889E-06	4.01	0.407E-08	4.98

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Table 5.4. Example 4. The $ P_k-P_{k-1}-[P_{k+2}]^2 $ element, on polygonal grids shown in Figure 5.4

$ k $	$ {{\mathcal T}}_l $	$ {{\|\hspace{-.02in}\|\hspace{-.02in}\|}} Q_hu-u_h{{\|\hspace{-.02in}\|\hspace{-.02in}\|}} $	Rate	$ \\|Q_hu-u_h\\| $	Rate
	6	0.3735E-01	1.00	0.7856E-04	2.00
1	7	0.1868E-01	1.00	0.1966E-04	2.00
	8	0.9339E-02	1.00	0.4916E-05	2.00
	5	0.1504E-02	1.98	0.3242E-05	2.95
2	6	0.3782E-03	1.99	0.4131E-06	2.97
	7	0.9482E-04	2.00	0.5216E-07	2.99
	4	0.1267E-03	2.97	0.4106E-06	3.95
3	5	0.1600E-04	2.99	0.2636E-07	3.96
	6	0.2010E-05	2.99	0.1673E-08	3.98
	2	0.5517E-03	3.98	0.5905E-05	5.26
4	3	0.3518E-04	3.97	0.1699E-06	5.12
	4	0.2234E-05	3.98	0.5253E-08	5.02

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