The Cell Boundary Element Method for Elliptic Pdes

We introduce the cell boundary element method(CBEM) for an elliptic PDE of the type:The PDE possesses no known fundamental solution for a general , therefore, we can not solve it by using a standard boundary integral equation method. But the PDE can be approximated by the Poisson equation locally in a small cell and the solution of the approximating Poisson equation can be represented by a boundary integral formula. Patching together each local solution via the local Dirichlet Neumann map by using the continuity of and the flux, ∇ on the boundary of each small cell, we have a global solution. The main advantage of our method is that the mesh generation is easy. Convergence of order is observed in -norm, where is the maximum diameter of celles