A Weak Galerkin Finite Element Method for Second-Order Elliptic Problems. (arXiv:1104.2897v1 [math.NA])

A Weak Galerkin Finite Element Method for Second-Order Elliptic Problems. (arXiv:1104.2897v1 [math.NA]): "

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 {\em 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.

"