Uma formulação Petrov-Galerkin descontínuo para solução da equação de Helmholtz com minimização do erro de fase

Abstract

Pollution error is a well known source of inaccuracies in continuous or discontinuous FE approaches to solve the Helmholtz equation. This topic is exaustivelly studied in a large number of papers as well as IHLENBURG e BABUSKA [1], IH- ˇ LENBURG [2] and others references inside there in and others references. Robust methodologies for structured square meshes have been developed in recent years. This work seeks to develop a methodology based on Discontinuous PetrovGalerkin formulation (DPG), in order to minimize phase error for structured or unstructured meshes applied for Helmholtz equation in homogeneous media. A Petrov–Galerkin FE formulation is introduced for Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the triangular mesh, a global basis function for the weighting space is obtained, adding to the bilinear C 0 Lagrangian weighting function linear combinations. The optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients α n m of these linear combinations attending to optimal criteria. This is done numerically through a preprocessing technique that is naturally applied to nonuniform 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 BABUSKA ˇ et al. [3] is obtained. Numerical results are presented illustrating the great stability and accuracy of this formulation with nonuniform and unstructured meshes.