A modified Galerkin FEM for 1D Helmholtz equations

A method is presented that aims to eliminate the numerical errors inherent in the standard Galerkin finite element method (GFEM) for solving homogeneous Helmholtz equations. An error analysis of the standard GFEM with linear elements is first performed by using the concept of truncation error in finite difference methods, and then the truncation error expression is obtained. A linear GFEM with an artificial stiffness is proposed to solve the Helmholtz equation after investigating the effect of the error on numerical solution. The proposed method is essentially as straightforward as the standard GFEM and thus requires almost no additional computational effort. Numerical results show that present pollution error decreases by 90% compared with that of the standard GFEM.