Analysis of a coupled finite-infinite element method for exterior Helmholtz problems

This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz problems. The key idea i s to represent both the exact and the numerical solution by the Dirichlet-t o-Neumann operators that they induce on the coupling hypersurface in the ex terior of an obstacle. The investigation of convergence can then be related to a spectral analysis of these DtN operators. We give a general outline o f our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere. Our main goal is to explore the convergen ce mechanism. In this context, we show well-posedness of both the continuou s and the discrete models. We further show that the discrete inf-sup consta nts have a positive lower bound that does not depend on the number of DOF o f the IEM. The proofs are based on lemmas on the spectra of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence.