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

For high wave numbers, the Helmholtz equation suffers the so-called 'pollut ion effect'. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Ga lerkin FEM is presented. This method does not require to compute the numeri cal solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p-method) is compared to modified methods presente d in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different 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 effective modifie d method is the QSFEM [1, 2] but it is also very complicated in the general setting. The residual-free bubble [3, 4] is effective in one dimension but not in higher dimensions. The least-square method [1, 5] approach lowers t he dispersion but relatively little. The results for triangular meshes show that the best topology is the 'criss-cross' pattern. Copyright (C) 1999 Jo hn Wiley & Sons, Ltd.