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)
Im. Babuska et Sa. Sauter, 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), SIAM REV, 42(3), 2000, pp. 451-484
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
.