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Error estimates for the ultra weak variational formulation of the Helmholtz equation. (English) Zbl 1155.65094

The ultra weak variational formulation (UWVF) of the Helmholtz equation provides a variational framework suitable for discretizations using plane waves solutions of an appropriate adjoint equation. Convergence of the method is only shown on the boundary of the domain. Substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. By using the fact that the UWVF is essentially an upwind discontinuous Galerkin method convergence is proved in the special case where there is no absorbing medium present.
Some other estimates are provided in the case when absorption is present, and some simple numerical results are presented to test the estimates. It is expected that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell’s equations and elasticity.

MSC:

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

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