A new approach for admissibility analysis of the direct discontinuous Galerkin method through Hilbert matrices

Abstract

This article continues the study of the so‐called direct discontinuous Galerkin (DDG) method for diffusion problems as developed in [Liu and Yan, SIAM J Numer Anal 47 (2009), 475–698;, Liu and Yan, Commun Comput Phys 8 (2010), 541–564; C. Vidden and J. Yan, J Comput Math 31 (2013), 638–662; H. Liu, Math Comp (in press)]. A key feature of the DDG method lies with the numerical flux design which includes two (or more) free parameters. This article identifies the class of all admissible numerical flux choices (Theorem 2.2) for degree n polynomial approximations for the symmetric DDG method [C. Vidden and J. Yan, J Comput Math 31 (2013), 638–662], guaranteeing stability of the resulting method. Our main contribution is the new technique of analysis for the DDG admissibility condition. The strategy is to directly evaluate the admissibility condition (Lemma 2.4) by choosing a simple polynomial basis. The admissibility condition is then transformed into an eigenvalue problem resulting in showing needed properties of inverse Hilbert matrices (Lemma 2.3, Appendix). Numerical tests are provided to confirm theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 350–367, 2016