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 a rectangular mesh with refined zones. The method consists in representing the unknown distribution and its derivatives by piecewise constant distributions (PCD) on distinct meshes together with an appropriate approximate variational formulation of the exact BVP on this piecewise constant distributions space. This method, named the PCD method, has the advantage of producing the most compact possible discrete stencil. Here, we analyze and prove the convergence of the PCD method and determine upper bounds on its convergence rate.

Keywords

• Boundary value problem
• PCD method
• local mesh refinement
• most compact discrete scheme
• interface boundary
• $O(h)$-convergence rate