A Class of Discontinuous Petrov-Galerkin Methods. Part IV: Wave Propagation

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for onedimensional time-harmonic wave propagation problems. The method is constructed within the framework of the Discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods pick solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm to achieve error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L-norm as the energy norm. We obtain uniform stability with respect to the wave number k, thus eliminating the phase error of the numerical solution. We illustrate the method with a number of 1D numerical experiments, using discontinuous (L-conforming) piecewise polynomials hp spaces for the trial space with the corresponding optimal test functions computed approximately at the element level. The 1D experiments are accompanied with a complete stability analysis.