The PCD Method on Composite Grid

Abstract

We introduce a discretization method of boundary value problems(BVP) in the case of the two dimensional diffusion equation on arectangular mesh with refined zones. The method consists inrepresenting the unknown distribution and its derivatives bypiecewise constant distributions (PCD) on distinct meshes togetherwith an appropriate approximate variational formulation of the exactBVP on this piecewise constant distributions space. This method,named the PCD method, has the advantage of producing the mostcompact possible discrete stencil. Here, we analyze and prove theconvergence of the PCD method and determine upper bounds on itsconvergence rate.