On mean value solutions for the Helmholtz equation on square grids Original Research Article

A numerical treatment for the boundary value problem involving the Helmholtz equation Δu−λ2u=f is presented. The method is a five-point formula with an improved accuracy when compared with the usual finite difference method. Besides, the accuracy evaluation is provided in analytical form and the classical difference scheme is seen as a truncated series approximation to the present method. The idea comes from approximations to analytical solutions to the Dirichlet problem inside a ball, based on the Green identity. The homogeneous and the nonhomogeneous parts are evaluated in separate expressions, and the precision error yielded is of order O(h2). Some numerical examples and comparisons are presented.