Local boundary element based a new finite difference representation for Poisson equations

We present a new finite difference method for solving Poisson's equation with the Dirichlet boundary condition on a more general type of discretization for given domain, based on the local boundary element method. The method uses the piecewise linear approximation and produce a sparse linear system despite the use of boundary elements. The discrete maximum principal is established without any angle condition for the discrete cells of the discretization. The convergence behavior is comparable to that of standard finite difference methods on rectangle grids, and equally super-convergence property is attained on more general meshes when the solution u is in the function class C ^2,α(Ω)∪C³(Ω), 0<α<1. Also, if u∈C^3,1(Ω), the standard O(h²) convergence is obtained. Numerical tests are given, which illustrate our results.