AN IMPROVEMENT IN THE GALERKIN PROJECTED RESIDUAL FINITE ELEMENT METHOD (GPR) FOR HELMHOLTZ EQUATION

Abstract. In XXVIII CILAMCE edition a new finite element method for Helmholtz equation was introduced: The Galerkin Plus Multiplies Projection of Residual Method (GMPR). This method was obtained adding to the Galerkin formulation an appropriate numbers of projections of the residual of PDE within each element. This allows that the element matrix has a maximum number of free parameters. Also, for rectangular domain, uniform mesh and bilinear elements, a methodology to choose these free parameters was presented. The criterion adopted to determine the free parameters consists of minimizing the phase error of the approximate solution. The GMPR method is a “variationally ” consistent finite element formulation and convergent for the homogeneous Helmholtz equation. In spite of everything, this initial version of the GMPR method is not necessarily convergent for all non homogeneous problems. Therefore, in this work we introduced a necessary modification to obtain a FEM of the GPR class (initially denoted by GMPR and now by GPR) with uniform convergence properties. The modification is introduced through a new projection term in the weak formulation: the residual gradient projection term. With this modification the GPR formulation remains “variationally ” consistent and uniform convergence properties ar