## 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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