A weak Galerkin finite element method for second-order elliptic problems

This paper introduces a finite element method by using a weakly defined gradient operator over generalized functions. The use of weak gradients and their approximations results in a new concept called discrete weak gradients   which is expected to play an important role in numerical methods for partial differential equations. This article intends to provide a general framework for managing differential operators on generalized functions. As a demonstrative example, the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical scheme is called a weak Galerkin (WG) finite element method. 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 H1H1 and L2L2 norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.