Adaptive numerical solution of a discontinuous Galerkin method for a Helmholtz problem in low-frequency regime

作者：Barrios Tomas P; Bustinza Rommel^*; Dominguez Victor

来源：Journal of Computational and Applied Mathematics, 2016, 300: 312-340.

DOI：10.1016/j.cam.2015.12.024

• LDG
• Helmholtz problem
• Indefinite bilinear forms

We develop an a posteriori error analysis for Helmholtz problem using the local discontinuous Galerkin (LDG for short) approach. For the sake of completeness, we give a description of the main a priori-results of this method. Indeed, under some assumptions on regularity of the solution of an adjoint problem, we prove that: (a) the corresponding indefinite discrete scheme is well posed; (b) the approach is convergent, with the expected convergence rates as long as the meshsize h is small enough. We give precise information on how small h has to be in terms of the size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the Laplacian. After that, we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number. We finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work.

• 出版日期2016-7