Numerical Investigation of the Conditioning for Plane Wave Discontinuous Galerkin Methods
We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.
The authors have been funded by the Austrian Science Fund (FWF) through the project P 29197-N32. The third author has also been funded by the FWF through the project F 65.
- 2.A. Buffa, P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. M2AN Math. Model. Numer. Anal. 42(6), 925–940 (2008)Google Scholar
- 6.C. Gittelson, Plane wave discontinuous Galerkin methods, Master’s thesis, SAM, ETH Zurich, Switzerland, 2008Google Scholar
- 7.C.J. Gittelson, R. Hiptmair, I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version. M2AN Math. Model. Numer. Anal. 43(2), 297–331 (2009)Google Scholar
- 13.J. Melenk, On generalized finite element methods, Ph.D. thesis, University of Maryland, 1995Google Scholar
- 15.G.W. Stewart, Matrix Algorithms. Vol. I: Basic Decompositions (Society for Industrial and Applied Mathematics, Philadelphia, 1998)Google Scholar