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KAUST-IAMCS Workshop on Multiscale Modeling, Advanced Discretization Techniques, and Simulation of Wave Propagation

Ignacio Muga, Pontifical Catholic University of Valparaíso (Chile)
Analysis of the DPG Method for the Helmholtz Equation in Multi-Dimensions

Abstract

The numerical solution of wave propagation problems at high frequencies has been recognized as an outstanding challenge in numerical analysis. In general, numerical methods for wave propagation are subject to the effect of pollution: increasing the frequency, while maintaining the approximation quality of the numerical discretization, results in a divergence of the computed result from the best approximation the discretization is capable of. One of the main consequences of this is the, so-called, phase error [2].

In the famous article [1], Babuška & Sauter predicted that, in two dimensions, it is impossible to eliminate the pollution effect completely. Although they restrict their analysis to a standard Bubnov-Galerkin FEM with bilinear elements in a structured mesh, the result is deeper and one might extrapolate it to other FEM configurations and to higher dimensions.

As a continuation of our research on DPG methods for wave propagation [3], we present a pollution-free method for a Helmholtz model problem in multi-dimensions. We give a full stability analysis showing quasi-optimality of the method with mesh-independent and wavenumber-independent constant. Although we can not mathematically discard the phase error, numerical results in 2D show that it is practically unobservable.

References

  1. I.M. Babuška and S.A. Sauter, Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers?, SIAM J. Numer. Anal., 34(6):2392-2423, .
  2. F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Volume 132 of Applied Mathematical Sciences, Springer-Verlag, New York, .
  3. J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V.M. Calo, A Class of Discontinuous Petrov-Galerkin Methods. Part IV: Wave Propagation, Journal of Computational Physics 230, 2406-2432, .