Sharp preasymptotic error bounds for the Helmholtz h-FEM
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In the analysis of the h-version of the finite-element method (FEM), with fixed polynomial degree p, applied to the Helmholtz equation with wavenumber k≫ 1, the asymptotic regime is when (hk)^p C_ sol is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here C_ sol is the norm of the Helmholtz solution operator, normalised so that C_ sol∼ k for nontrapping problems. The preasymptotic regime is when (hk)^2pC_ sol is sufficiently small, and (for physical data) one expects the relative error of the Galerkin solution to be controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition approximated either by a radial perfectly-matched layer (PML) or an impedance boundary condition. Previously, such bounds for p>1 were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalisation of the "elliptic-projection" argument (the argument used to obtain the result for p=1) which can be applied to a wide variety of abstract Helmholtz-type problems.
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