How We solve PDEs

A finite difference method on an unstructured finite element mesh which we call finite difference element method (FDEM) is presented. The FDEM program package will be a black-box solver for nonlinear systems of elliptic and parabolic PDEs with mesh refinement and automatic control of the consistency order in each space grid point. In this paper we present the solution method (with examples) for 2-D systems of elliptic PDEs.