A Weak Galerkin Finite Element Method with Polynomial Reduction

The novel idea of weak Galerkin (WG) finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use second order elliptic problems to demonstrate the basic idea of polynomial reduction. A new weak Galerkin finite element method is proposed and analyzed. This new finite element scheme features piecewise polynomials of degree $k\ge 1$ on each element plus piecewise polynomials of degree $k-1\ge 0$ on the edge or face of each element. 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. In addition, the paper presents a great deal of numerical experiments to demonstrate the power of the WG method in dealing with finite element partitions consisting of arbitrary polygons in two dimensional spaces or polyhedra in three dimensional spaces. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honey comb mesh in 2d and mesh with deformed cubes in 3d. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.