A new fourth order discretization for singularly perturbed two dimensional non-linear elliptic boundary value problems

In this article, we derive a new difference method of O(h4), so called, arithmetic average discretization for the solution of two dimensional non-linear singularly perturbed elliptic partial differential equation of the form ε(uxx + uyy) = f(x, y, u, ux, uy), 0 < x, y < 1, subject to appropriate Dirichlet boundary conditions where ε > 0 is a small parameter .We also derive new methods of O(h4) for the estimates of (∂u/∂n), which are quite often of interest in many physical problems. In all cases, we require only 9-grid points and a single computational cell. The main advantage of the proposed methods is that the methods are directly applicable to singular problems. We do not require any special technique or modification to solve singular problems. Numerical results are provided to demonstrate the usefulness of the methods discussed.