A new weak Galerkin finite element scheme for general second-order elliptic problems

A new weak Galerkin (WG) finite element scheme is presented for general second-order elliptic problems in this paper. In this new scheme, a skew symmetric form has been used for handling the convection term. The advantage of the new scheme is that the system of linear equations from the scheme is positive definite and one might easily get the well-posedness of the system. In this scheme, the WG elements are designed to have the form of (Pk(T),Pk−1(e)). That is, we choose the polynomials of degree k≥1 on each element and the polynomials of degree k−1 on the edge/face of each element. As a result, fewer degrees of freedom are generated in the new WG finite element scheme. It is also worth pointing out that the WG finite element scheme is established on finite element partitions consisting of arbitrary shape of polygons/polyhedra which are shape regular. Optimal-order error estimates are presented for the corresponding numerical approximation in various norms. Some numerical results are reported to confirm the theory.