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 - <p style="text-align: justify;">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>