A Note on Optimal Spectral Bounds for Nonoverlapping Domain Decomposition Preconditioners for $hp$-Version Discontinuous Galerkin Methods
Year: 2016
International Journal of Numerical Analysis and Modeling, Vol. 13 (2016), Iss. 4 : pp. 513–524
Abstract
In this article, we consider the derivation of $hp$-optimal spectral bounds for a class of domain decomposition preconditioners based on the Schwarz framework for discontinuous Galerkin finite element approximations of second-order elliptic partial differential equations. In particular, we improve the bounds derived in our earlier article [P.F. Antonietti and P. Houston, J. Sci. Comput., 46(1):124-149, 2011] in the sense that the resulting bound on the condition number of the preconditioned system is not only explicit with respect to the coarse and fine mesh sizes $H$ and $h$, respectively, and the fine mesh polynomial degree $p$, but now also explicit with respect to the polynomial degree $q$ employed for the coarse grid solver. More precisely, we show that the resulting spectral bounds are of order $p^{2}H/(qh)$ for the $hp$-version of the discontinuous Galerkin method.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2016-IJNAM-450
International Journal of Numerical Analysis and Modeling, Vol. 13 (2016), Iss. 4 : pp. 513–524
Published online: 2016-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: Schwarz preconditioners $hp$-discontinuous Galerkin methods.