A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for $hp$--version discontinuous Galerkin methods

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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Paola Antonietti, Paul Houston, Iain Smears. A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for $hp$--version discontinuous Galerkin methods. International Journal of Numerical Analysis and Modeling, Institute for Scientific Computing and Information, 2016, 13 (4), pp.513-524. ⟨hal-01428749⟩

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