Reliability of finite element methods for the numerical computation of waves

摘要

The numerical computation of stationary waves in exteriour and scattering problems is based on indefinite variational forms connected with the Helmholtz equation triangle openu+k~2u=0 where k is a real parameter (scalar wavenumber). Unlike the case of linear elasticity, stable dependence of both analytical (if existing) and numerical solutions on the data is not straightforward. Stability estimates of the form ||u||i<=C_(ij)||f||_j do hold for various norms i,j but the constants C_(ij) depend in general on the parameter k. Hence also the quality of the discrete solution depends on k, as well as on the parameters of the numerical model (stepwidth h, degree of approximation p). For practical application it is essential to have reliable "rules of the thumb" for the choice of the numerical parameters as a function of physical parameters.