A fourth-order Magnus scheme for Helmholtz equation
Journal of Computational and Applied Mathematics
Journal of Computational Physics
An optimal 25-point finite difference scheme for the Helmholtz equation with PML
Journal of Computational and Applied Mathematics
Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the $p$-Version
SIAM Journal on Numerical Analysis
Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation
SIAM Journal on Numerical Analysis
A finite element method enriched for wave propagation problems
Computers and Structures
A sweeping preconditioner for time-harmonic Maxwell's equations with finite elements
Journal of Computational Physics
Journal of Computational Physics
Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number
Journal of Computational Physics
Performance of an implicit time integration scheme in the analysis of wave propagations
Computers and Structures
Journal of Computational Physics
An explicit time integration scheme for the analysis of wave propagations
Computers and Structures
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The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of robustness by various modifications of the classical Galerkin FEM.However, we will prove that, in two and more space dimensions, it is impossible to eliminate this so-called pollution effect. In contrast, we will present a generalized FEM in one dimension that behaves robustly (i.e., is pollution-free) with respect to the wave number.The theory developed in this paper can also be used for the comparison of different discretization methods with respect to the size of their pollution.