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Finite element solution of the Helmholtz equation with high wave number. I: The \(h\)-version of the FEM. (English) Zbl 0838.65108

This paper examines the quality of the discrete numerical solutions to the Helmholtz equation \(\Delta u + k^2u = f\) where \(k\) is the wave number. These equations arise in problems of wave scattering and fluid-solid-interaction.
It is shown that the relative error in the finite element solution in \(H^1\) seminorm is \[ e_1 \leq C_1kh + C_2 k^3 h^2 \] where \(h\) is the step length of the meshes. The first term on the right hand side of the inequality is the approximation error and the second term is due to numerical pollution.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N15 Error bounds for boundary value problems involving PDEs
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