Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? (Reprinted from SIAM Journal Numerical Analysis, vol 32, pg 2392-2423, 1997)

The development of numerical methods for solving the Helmholtz equation, wh ich behaves robustly with respect to the wave number, is a topic of vivid r esearch. It was observed that the solution of the Galerkin finite element m ethod (FEM) differs significantly from the best approximation with increasi ng wave number. Many attempts have been presented in the literature to elim inate this lack of robustness by various modifications of the classical Gal erkin FEM. However, we will prove that, in two and more space dimensions, it is imposs ible to eliminate this so-called pollution effect. In contrast, we will pre sent a generalized FEM in one dimension that behaves robustly (i.e., is pol lution-free) with respect to the wave number. The theory developed in this paper can also be used for the comparison of d ifferent discretization methods with respect to the size of their pollution .