Is the Pollution Effect of the FEM Avoidable for the Helmholtz EquationConsidering High Wave Numbers

摘要

Using the Galerkin FEM to solve the Helmholtz equation, the error of thecorresponding solution differs from the error of the best approximation substantially, and this effect increases with higher wave numbers k. In the one dimensional case, we will define a stabilized variant of the FEM which fits in the setting of the generalized FEM (GFEM). This variant will produce a FE-solution being very close to the best approximation, and we can prove quasioptimal error estimates without any pollution effect. This situation changes essentially in the higher dimensional case. We will show that for every GFEM there exists a domain and a solution of the Helmholtz equation which cannot be approximated without any pollution. Numerical examples will illustrate the improvement of the stabilized FEM compared with the Galerkin FEM.