# American Institute of Mathematical Sciences

• Previous Article
Complex dynamics of a SIRS epidemic model with the influence of hospital bed number
• DCDS-B Home
• This Issue
• Next Article
Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters

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

 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

Received  April 2020 Revised  August 2020 Published  September 2020

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

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.

Citation: Xiu Ye, Shangyou Zhang. A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020277
##### References:

show all references

##### References:
Example 1. The first three triangular grids
Example 2. The first three triangular grids
Example 3. The first three rectangular grids
Example 4. The first three polygonal grids
Weak gradient calculated by (1.4), ${{|||}}\cdot{{|||}} = O(h^{r_1})$ and $\|\cdot\| = O(h^{r_2})$
 $P_{k}(T)$ $P_{k-1}(e)$ $[P_{k+1}(T)]^d$ $r_1$ $r_2$ $P_1(T)$ $P_0(e)$ $[P_2(T)]^2$ $0$ $0$ $P_2(T)$ $P_1(e)$ $[P_3(T)]^2$ $1$ $2$ $P_3(T)$ $P_2(e)$ $[P_4(T)]^2$ $2$ $3$
 $P_{k}(T)$ $P_{k-1}(e)$ $[P_{k+1}(T)]^d$ $r_1$ $r_2$ $P_1(T)$ $P_0(e)$ $[P_2(T)]^2$ $0$ $0$ $P_2(T)$ $P_1(e)$ $[P_3(T)]^2$ $1$ $2$ $P_3(T)$ $P_2(e)$ $[P_4(T)]^2$ $2$ $3$
Weak gradient calculated by (2.3), ${{|||}}\cdot{{|||}} = O(h^{r_1})$ and $\|\cdot\| = O(h^{r_2})$
 $P_{k}(T)$ $P_{k-1}(e)$ $[P_{k+1}(T)]^d$ $r_1$ $r_2$ $P_1(T)$ $P_0(e)$ $[P_2(T)]^2$ $1$ $2$ $P_2(T)$ $P_1(e)$ $[P_3(T)]^2$ $2$ $3$ $P_3(T)$ $P_2(e)$ $[P_4(T)]^2$ $3$ $4$
 $P_{k}(T)$ $P_{k-1}(e)$ $[P_{k+1}(T)]^d$ $r_1$ $r_2$ $P_1(T)$ $P_0(e)$ $[P_2(T)]^2$ $1$ $2$ $P_2(T)$ $P_1(e)$ $[P_3(T)]^2$ $2$ $3$ $P_3(T)$ $P_2(e)$ $[P_4(T)]^2$ $3$ $4$
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
 $k$ ${{\mathcal T}}_l$ ${{|||}} Q_hu-u_h{{|||}}$ 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
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
 $k$ ${{\mathcal T}}_l$ ${{|||}} Q_hu-u_h{{|||}}$ 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
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
 $k$ ${{\mathcal T}}_l$ ${{|||}} Q_hu-u_h{{|||}}$ 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
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
 $k$ ${{\mathcal T}}_l$ ${{|\hspace{-.02in}|\hspace{-.02in}|}} Q_hu-u_h{{|\hspace{-.02in}|\hspace{-.02in}|}}$ 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
 [1] Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 [2] Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 [3] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [4] Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 [5] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [6] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [7] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [8] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [9] Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230 [10] Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 [11] Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127 [12] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [13] Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 [14] Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115 [15] Waixiang Cao, Lueling Jia, Zhimin Zhang. A $C^1$ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327 [16] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 [17] Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 [18] José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 [19] Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 [20] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables