Volume 13, Issue 4
A Note on Optimal Spectral Bounds for Nonoverlapping Domain Decomposition Preconditioners for $hp$-Version Discontinuous Galerkin Methods

Int. J. Numer. Anal. Mod., 13 (2016), pp. 513-524.

Published online: 2016-07

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• 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.

• Keywords

Schwarz preconditioners, $hp$-discontinuous Galerkin methods.

65N30, 65N55, 65F08

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@Article{IJNAM-13-513, author = {}, title = {A Note on Optimal Spectral Bounds for Nonoverlapping Domain Decomposition Preconditioners for $hp$-Version Discontinuous Galerkin Methods}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {4}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/450.html} }
TY - JOUR T1 - A Note on Optimal Spectral Bounds for Nonoverlapping Domain Decomposition Preconditioners for $hp$-Version Discontinuous Galerkin Methods JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 513 EP - 524 PY - 2016 DA - 2016/07 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/450.html KW - Schwarz preconditioners, $hp$-discontinuous Galerkin methods. AB -

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.

P. F. Antonietti, P. Houston & I. Smears. (1970). A Note on Optimal Spectral Bounds for Nonoverlapping Domain Decomposition Preconditioners for $hp$-Version Discontinuous Galerkin Methods. International Journal of Numerical Analysis and Modeling. 13 (4). 513-524. doi:
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