# American Institute of Mathematical Sciences

• Previous Article
A diffusive weak Allee effect model with U-shaped emigration and matrix hostility
• 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] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [9] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377 [10] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [11] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [12] Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 [13] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [14] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [15] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268 [16] Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356 [17] Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $\Lambda$-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328 [18] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 [19] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [20] Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

2019 Impact Factor: 1.27