Smears, I;
(2018)
Nonoverlapping Domain Decomposition Preconditioners for Discontinuous Galerkin Approximations of Hamilton–Jacobi–Bellman Equations.
Journal of Scientific Computing
, 74
(1)
pp. 145-174.
10.1007/s10915-017-0428-5.
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Abstract
We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximations of fully nonlinear Hamilton–Jacobi–Bellman (HJB) partial differential equations. These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix A. In this work, we construct a nonoverlapping domain decomposition preconditioner P, that is based on A, and we then show that the effectiveness of the preconditioner for solving the nonsymmetric problems can be studied in terms of the condition number κ(P−1A). In particular, we establish the bound κ(P−1A)≲1+p6H3/q3h3, 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 such result for this class of methods that explicitly accounts for the dependence of the condition number on q; our analysis is founded upon an original optimal order approximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these methods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under h-refinement for moderate polynomial degrees.
| Type: | Article |
|---|---|
| Title: | Nonoverlapping Domain Decomposition Preconditioners for Discontinuous Galerkin Approximations of Hamilton–Jacobi–Bellman Equations |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s10915-017-0428-5 |
| Publisher version: | http://dx.doi.org/10.1007/s10915-017-0428-5 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Domain decomposition, Preconditioners, GMRES, Discontinuous Galerkin, Finite element methods, Approximation in discontinuous spaces, Hamilton–Jacobi–Bellman equations |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1572495 |
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