A NUMERICAL COMPARISON OF FINITE ELEMENT METHODS FOR THE HELMHOLTZ EQUATION

Three finite element formulations for the solution of the Helmholtz equation are considered. The performance of these methods is compared by performing a discrete dispersion analysis and by solving two canonical problems on nonuniform meshes. It is found that: (1) The scaled L₂ error for the Galerkin method, using linear interpolation functions, grows as k(kh)², indicating the pollution inherent in this method; (2) The Galerkin least squares method is more accurate, but does display significant pollution error; (3) The residual-based method of Oberai & Pinsky,⁸ which was designed to be almost pollution-free for uniform meshes retains its accuracy on nonuniform meshes; (4) The computational cost of implementing all these formulations is approximately the same.