A quasi optimal Petrov–Galerkin method for Helmholtz problem

Abstract

AbstractA Petrov–Galerkin finite element formulation is introduced for Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the mesh, a global basis function for the weighting space is obtained adding to the bilinear C0 Lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node. Quasi optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimality criteria. This is done numerically through a simple preprocessing technique naturally applied to non‐uniform and unstructured meshes. In particular, for uniform mesh a quasi optimal interior stencil of the same order of the quasi‐stabilized finite element method stencil derived by Babuška et al. (Comput. Methods Appl. Mech. Engrg 1995; 128:325–359) is obtained. Numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non‐uniform meshes. Copyright © 2009 John Wiley & Sons, Ltd.