A Conservative Finite Difference Scheme for Static Diffusion Equation

A new discretization scheme for partial differential equations, based on the finite differences method, and its application to the two dimensional static diffusion equation is presented. This scheme produces better approximations than a standard use of finite differences. It satisfies properties of continuous differential operators and discrete versions of integral identities, which guarantee its conservative character. In addition, a global quadratic convergence rate is obtained naturally from the gradient second order one-sided approximation at the boundary nodes on a non-uniform staggered grid (of distributed points). This approach avoids the commonly used ghost points or extended grid concepts.