Asymptotic anaylsis of a singular perturbation problem
SIAM Journal on Mathematical Analysis
Journal of Computational Physics
A numerical scheme based on mean value solutions for the Helmholtz equation on triangular grids
Mathematics of Computation
Matrix computations (3rd ed.)
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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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.