A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D
Demkowicz, Leszek F.
Calo, Victor M.
KAUST DepartmentApplied Mathematics and Computational Science Program
Physical Sciences and Engineering (PSE) Division
Environmental Science and Engineering Program
Numerical Porous Media SRI Center (NumPor)
MetadataShow full item record
AbstractThe 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 one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. 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 select 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 that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed. © 2010 Elsevier Inc.
SponsorsJ. Zitelli was supported by an ONR Graduate Traineeship and CAM Fellowhip. I. Muga was supported by Sistema Bicentenario BECAS CHILE (Chilean Government). L. Demkowicz was supported by a Collaborative Research Grant from King Abdullah University of Science and Technology (KAUST). J. Gopalakrishnan was supported by the National Science Foundation under Grant No. DMS-1014817.
JournalJournal of Computational Physics