Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions

For high wave numbers, the Helmholtz equation su!ers the so-called &pollution e!ectʹ. This e!ect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p-method) is compared to modi"ed methods presented in the literature. A study of the in#uence of the topology triangles is also carried out. The e$ciency of the di!erent methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most e!ective modi"ed method is the QSFEM [1, 2] but it is also very complicated in the general setting. The residual-free bubble [3, 4] is e!ective in one dimension but not in higher dimensions. The least-square method [1, 5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the &criss-crossʹ pattern. Copyright ( 1999 John Wiley & Sons, Ltd.