In this paper, we design some efficient domain decomposition preconditioners for the discontinuous Petrov–Galerkin (DPG) method. Due to the special properties of the DPG method, the boundary condition becomes crucial in both of its application and analysis. We mainly focus on one of the boundary conditions: the Robin boundary condition, which actually appears in some useful model problems like the Helmholtz equation. We first design a two-level additive Schwarz preconditioner for the Poisson equation with a Robin boundary condition and give a rigorous condition number estimate for the preconditioned algebraic system. Moreover we also construct an additive Schwarz preconditioner for solving the Helmholtz equation. Numerical results show that the condition number of the preconditioned system is independent of wavenumber and mesh size .
Accepté le :
DOI : 10.1051/m2an/2016050
Mots clés : DPG, domain decomposition, additive Schwarz preconditioner, Robin boundary condition, Helmholtz equation
@article{M2AN_2017__51_3_1021_0, author = {Li, Xiang and Xu, Xuejun}, title = {Domain decomposition preconditioners for the discontinuous {Petrov{\textendash}Galerkin} method}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1021--1044}, publisher = {EDP-Sciences}, volume = {51}, number = {3}, year = {2017}, doi = {10.1051/m2an/2016050}, mrnumber = {3666655}, zbl = {1373.65084}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016050/} }
TY - JOUR AU - Li, Xiang AU - Xu, Xuejun TI - Domain decomposition preconditioners for the discontinuous Petrov–Galerkin method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1021 EP - 1044 VL - 51 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016050/ DO - 10.1051/m2an/2016050 LA - en ID - M2AN_2017__51_3_1021_0 ER -
%0 Journal Article %A Li, Xiang %A Xu, Xuejun %T Domain decomposition preconditioners for the discontinuous Petrov–Galerkin method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1021-1044 %V 51 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016050/ %R 10.1051/m2an/2016050 %G en %F M2AN_2017__51_3_1021_0
Li, Xiang; Xu, Xuejun. Domain decomposition preconditioners for the discontinuous Petrov–Galerkin method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1021-1044. doi : 10.1051/m2an/2016050. http://www.numdam.org/articles/10.1051/m2an/2016050/
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