Applied Numerical Mathematics 41 (2002) 459–479 www.elsevier.com/locate/apnum On mean value solutions for the Helmholtz equation on square grids ✩ João B.R. do Val a,∗ , Marinho G. Andrade b a UNICAMP—Universidade de Campinas, Fac. de Eng. Elétrica e de Computação, Depto. de Telemática, C.P. 6101, 13081-970 Campinas, SP, Brazil b USP—Universidade de São Paulo, Inst. de Ciências Matemáticas e Computação, 13560-970 São Carlos, SP, Brazil Abstract A numerical treatment for the boundary value problem involving the Helmholtz equation u - λ 2 u = 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(h 2 ). Some numerical examples and comparisons are presented.  2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Numerical approximation; Finite difference method; Partial differential equation; Helmholtz equation 1. Introduction The finite difference method is one of the standard numerical methods for boundary value problems, and it is usually obtained by direct discretization of the differential operator. The corresponding truncation via Taylor series provides the bound for the precision error. In this paper we consider numerical solutions on square mesh grids for the Helmholtz equation u - λ 2 u = f , that are not based on the usual Euler discretization, but are obtained by a method called Mean Value Scheme (MVS), cf. [2]. The MVS is based on an approximation for integrals on the circle associated with the solution of the Helmholtz equation in the interior of this particular domain. The approximation is based on a result that can be seen as an extension of the mean value theorem for harmonic functions for the Helmholtz equation, which explain the given name. Any circle to be considered is adjusted to contain the four neighboring points to ✩ This work was partially supported by CNPq under Grant N. 300721/86-2(NV). * Corresponding author. E-mail address: jbosco@dt.fee.unicamp.br (J.B.R. do Val). 0168-9274/01/$22.00  2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0168-9274(01)00127-1