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## 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
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##### 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
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