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In this paper, authors shall introduce a finite element method by using aweakly defined gradient operator over discontinuous functions withheterogeneous properties. The use of weak gradients and their approximationsresults in a new concept called {\em discrete weak gradients} which is expectedto play important roles in numerical methods for partial differentialequations. This article intends to provide a general framework for operatingdifferential operators on functions with heterogeneous properties. As ademonstrative example, the discrete weak gradient operator is employed as abuilding block to approximate the solution of a model second order ellipticproblem, in which the classical gradient operator is replaced by the discreteweak gradient. The resulting numerical approximation is called a weak Galerkin(WG) finite element solution. It can be seen that the weak Galerkin methodallows the use of totally discontinuous functions in the finite elementprocedure. For the second order elliptic problem, an optimal order errorestimate in both a discrete $H^1$ and $L^2$ norms are established for thecorresponding weak Galerkin finite element solutions. A superconvergence isalso observed for the weak Galerkin approximation.

A Weak Galerkin Finite Element Method for Second-Order Elliptic Problems