Abstract
AbstractA quasi optimal finite difference method (QOFD) is proposed for the Helmholtz problem. The stencils' coefficients are obtained numerically by minimizing a least‐squares functional of the local truncation error for plane wave solutions in any direction. In one dimension this approach leads to a nodally exact scheme, with no truncation error, for uniform or non‐uniform meshes. In two dimensions, when applied to a uniform cartesian grid, a 9‐point sixth‐order scheme is derived with the same truncation error of the quasi‐stabilized finite element method (QSFEM) introduced by Babuška et al. (Comp. Meth. Appl. Mech. Eng. 1995; 128:325–359). Similarly, a 27‐point sixth‐order stencil is derived in three dimensions. The QOFD formulation, proposed here, is naturally applied on uniform, non‐uniform and unstructured meshes in any dimension. Numerical results are presented showing optimal rates of convergence and reduced pollution effects for large values of the wave number. Copyright © 2009 John Wiley & Sons, Ltd.
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