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Abstract
1- Introduction
2- Weak Galerkin finite element schemes
3- Well posedness
4- Error estimates in energy norm
5- Error estimates in L 2 norm
6- Numerical experiments
References
Abstract
A stabilizing/penalty term is often used in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous Galerkin finite element methods will simplify formulations and reduce programming complexity significantly. The goal of this paper is to introduce a stabilizer free weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This new WG method keeps a simple symmetric positive definite form and can work on polygonal/polyhedral meshes. Optimal order error estimates are established for the corresponding WG approximations in both a discrete H 1 norm and the L 2 norm. Numerical results are presented verifying the theorem.
6. Numerical experiments
We solve the following Poisson equation on the unit square:
− ∆u = 2π 2 sin πx sin πy, (x, y) ∈ Ω = (0, 1)2 , (6.1)
with the boundary condition u = 0 on ∂Ω. In the first computation, the level one grid consists of two unit right triangles cutting from the unit square by a forward slash. The high level grids are the half-size refinements of the previous grid. The first three levels of grids are plotted in Fig. 6.1. The error and the order of convergence are shown in Table 6.1. The numerical results confirm the convergence theory. In Fig. 6.2, we plot the finite element solution and the discretization errors on triangular and on polygonal grids. We can see, with same number of unknowns, the solutions on triangular grids are more accurate than those on polygonal grids. This can also be seen from the two data tables. In the next computation, we use a family of polygonal grids (with 12-side polygons) shown in Fig. 6.3. The numerical results in Table 6.2 indicate that the polynomial degree j for the weak gradient needs to be larger, which confirms the theory: j depending on the number of edges of a polygon. The convergence history confirms the theory.