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Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? (English) Zbl 0956.65095
It is proved that a pollution-free modification of the Galerkin method can be constructed in the framework of generalized finite element methods (FEMs), in the one-dimensional case. In two or more dimensions, a substantial reduction in pollution effect can be achieved (resulting in a quasi-stabilized FEM), though a complete removal of the effect is not possible.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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