This work was supported in part by the AFOSR under FA9550-09-1-0608, the NSF under grant DMS-1014817, the FONDECYT project 1110272, and an ONR Graduate Traineeship.
Computer Methods in Applied Mechanics & Engineering
Helmholtz equation -- Numerical solutions, Nonstandard mathematical analysis, Sound-waves, Polynomials
We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with several other recent works that gave wave number explicit estimates. Numerically, comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures. The new DPG method is designed following the previously established principles of optimal test functions. In addition to the nonstandard test functions, in this work, we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates.
Demkowicz, Leszek; Gopalakrishnan, Jay; Muga, Ignacio; and Zitelli, Jeffrey, "Wavenumber Explicit Analysis of a DPG Method for the Multidimensional Helmholtz Equation" (2012). Mathematics and Statistics Faculty Publications and Presentations. 39.