A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Abstract

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form $P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2$, where $k\ge 1$ is the degree of polynomials in the interior of the element T, $j\ge 0$ is the degree of polynomials on the boundary of T, and $\ell \ge 0$ is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.

Appendix

In this section we present some detailed computational data for a set of selected values of $k,\ j$, and $\ell$. These data shall be in support of the order of convergence reported in Sect. 5. The numerical results are organized as follows. Tables 9, 10, 11, 12, 13, and 14 illustrate the table index number for the set value of $(k,j,\ell )$, and the rest of the tables show the corresponding numerical results. For example, Table 9 points to the table index number when the stabilizer $j_\partial (v) = Q_b^m(v_0-v_b)$ was employed in the numerical scheme. This table has a fixed value of $k=2$ while j and $\ell$ are varying. The entry of the table at $(k,j,\ell )=(2,1,2)$ has value (Table 17) so that the computational results for $(k,j,\ell )=(2,1,2)$ should be found in Table 17.

1.1 Index Tables

The index tables are given in Tables 9, 10, 11, 12, 13, and 14. Please be reminded that the values in those tables refer to the table number where the computational results are reported.

Table 9 Table index for $j_\partial (v) = Q_b^m(v_0-v_b)$ with $k=2$

Table 10 Table index for $j_\partial (v) = Q_b^m(v_0-v_b)$ with $k=3$

Table 11 Table index for $j_\partial (v) = Q_b^m(v_0-v_b)$ with $k=4$

Table 12 Table index for $j_\partial (v) = v_0-v_b$ with $k=2$

Table 13 Table Index for $j_\partial (v) = v_0-v_b$ with $k=3$

Table 14 Table index for $j_\partial (v) = v_0-v_b$ with $k=4$

1.2 Tables for Computational Results

All the detailed computational results are presented in Tables 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, and 42. No interpretation of the data is necessary as they are virtually self-explanatory. Interested readers are invited to draw their own conclusions from reading these numerical results.

Table 15 Convergence orders for $k=2$, $j=0$, $\ell =1$

Table 16 Convergence orders for $k=2$, $j=1$, $\ell =1$

Table 17 Convergence orders for $k=2$, $j=1$, $\ell =2$

Table 18 Convergence orders for $k=2$, $j=2$, $\ell =2$

Table 19 Convergence orders for $k=2$, $j=3$, $\ell =3$

Table 20 Convergence orders for $k=2$, $j=4$, $\ell =4$

Table 21 Convergence orders for $k=3$, $j=1$, $\ell =2$

Table 22 Convergence orders for $k=3$, $j=2$, $\ell =2$

Table 23 Convergence orders for $k=3$, $j=2$, $\ell =3$

Table 24 Convergence orders for $k=3$, $j=3$, $\ell =3$

Table 25 Convergence orders for $k=3$, $j=4$, $\ell =4$

Table 26 Convergence orders for $k=4$, $j=1$, $\ell =3$

Table 27 Convergence orders for $k=4$, $j=2$, $\ell =3$

Table 28 Convergence orders for $k=4$, $j=3$, $\ell =3$

Table 29 Convergence orders for $k=4$, $j=3$, $\ell =4$

Table 30 Convergence orders for $k=4$, $j=4$, $\ell =4$

Table 31 Convergence orders for $k=2$, $j=1$, $\ell =1$

Table 32 Convergence orders for $k=2$, $j=2$, $\ell =2$

Table 33 Convergence orders for $k=2$, $j=3$, $\ell =3$

Table 34 Convergence orders for $k=2$, $j=4$, $\ell =4$

Table 35 Convergence orders for $k=3$, $j=1$, $\ell =1$

Table 36 Convergence orders for $k=3$, $j=2$, $\ell =2$

Table 37 Convergence orders for $k=3$, $j=3$, $\ell =3$

Table 38 Convergence orders for $k=3$, $j=4$, $\ell =4$

Table 39 Convergence orders for $k=4$, $j=1$, $\ell =2$

Table 40 Convergence orders for $k=4$, $j=2$, $\ell =2$

Table 41 Convergence orders for $k=4$, $j=3$, $\ell =3$

Table 42 Convergence orders for $k=4$, $j=4$, $\ell =4$