A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H 1 and L 2 norms are established for the corresponding weak Galerkin mixed finite element solutions.Key words. weak Galerkin, finite element methods, discrete weak divergence, second-order elliptic problems, mixed finite element methods

In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play important roles in numerical methods for partial differential equations. This article intends to provide a general framework for operating differential operators on functions with heterogeneous properties. As a demonstrative example, the discrete weak gradient operator is employed as a building block to approximate the solution of a model second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical approximation is called a weak Galerkin (WG) finite element solution. It can be seen that the weak Galerkin method allows the use of totally discontinuous functions in the finite element procedure. For the second order elliptic problem, an optimal order error estimate in both a discrete H 1 and L 2 norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.

Abstract. A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete H 2 norm is established for the corresponding WG finite element solutions. Error estimates in the usual L 2 norm are also derived, yielding a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions.

Abstract. This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥ 1 for the velocity and polynomials of degree k − 1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k − 1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.