A stabilized discontinuous finite element method for elliptic problems

A new finite element method is proposed and analysed for second order elliptic equations using discontinuous piecewise polynomials on a finite element partition consisting of general polygons. The new method is based on a stabilization of the well‐known primal hybrid formulation by using some least‐squares forms imposed on the boundary of each element. Two finite element schemes are presented. The first one is a non‐symmetric formulation and is absolutely stable in the sense that no parameter selection is necessary for the scheme to converge. The second one is a symmetric formulation, but is conditionally stable in that a parameter has to be selected in order to have an optimal order of convergence. Optimal‐order error estimates in some H 1 ‐equivalence norms are established for the proposed discontinuous finite element methods. For the symmetric formulation, an optimal‐order error estimate is also derived in the L 2 norm. The new method features a finite element partition consisting of general polygons as opposed to triangles or quadrilaterals in the standard finite element Galerkin method.