The Mathematical Institute, University of Oxford, Eprints Archive

Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin finite element methods in $H^2$-type norms

Smears, Iain (2013) Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin finite element methods in $H^2$-type norms. Technical Report. Unspecified. (Submitted)

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Abstract

We analyse the spectral bounds of nonoverlapping domain decomposition preconditioners for $hp$-version discontinuous Galerkin finite element methods in $H^2$-type norms, which arise in applications to fully nonlinear Hamilton--Jacobi--Bellman partial differential equations. We show that for a symmetric model problem, the condition number of the preconditioned system is at most of order $1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first result for this class of methods that explicitly accounts for the dependence of the condition number on $q$, and its sharpness is shown numerically. The key analytical tool is an original optimal order approximation result between fine and coarse discontinuous finite element spaces.

We then go beyond the model problem and show computationally that these methods lead to efficient and competitive solvers in practical applications to nonsymmetric, fully nonlinear Hamilton--Jacobi--Bellman equations.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1711
Deposited By: Lotti Ekert
Deposited On:11 Jul 2013 07:58
Last Modified:29 May 2015 19:24

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