Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization: The many-subdomain case

Gander, Martin; Hajian, Soheil

doi:10.1090/mcom/3293

Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization: The many-subdomain case
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by Martin J. Gander and Soheil Hajian;

Math. Comp. 87 (2018), 1635-1657

DOI: https://doi.org/10.1090/mcom/3293

Published electronically: September 29, 2017

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Abstract:

Schwarz methods are attractive parallel solution techniques for solving large-scale linear systems obtained from discretizations of partial differential equations (PDEs). Due to the iterative nature of Schwarz methods, convergence rates are an important criterion to quantify their performance. Optimized Schwarz methods (OSM) form a class of Schwarz methods that are designed to achieve faster convergence rates by employing optimized transmission conditions between subdomains. It has been shown recently that for a two-subdomain case, OSM is a natural solver for hybridizable discontinuous Galerkin (HDG) discretizations of elliptic PDEs. In this paper, we generalize the preceding result to the many-subdomain case and obtain sharp convergence rates with respect to the mesh size and polynomial degree, the subdomain diameter, and the zeroth-order term of the underlying PDE, which allows us for the first time to give precise convergence estimates for OSM used to solve parabolic problems by implicit time stepping. We illustrate our theoretical results with numerical experiments.

References

Paola F. Antonietti and Blanca Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case, M2AN Math. Model. Numer. Anal. 41 (2007), no. 1, 21–54. MR 2323689, DOI 10.1051/m2an:2007006
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal. 41 (2003), no. 1, 306–324. MR 1974504, DOI 10.1137/S0036142902401311
Xiao-Chuan Cai, Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer. Math. 60 (1991), no. 1, 41–61. MR 1131498, DOI 10.1007/BF01385713
Paul Castillo, Performance of discontinuous Galerkin methods for elliptic PDEs, SIAM J. Sci. Comput. 24 (2002), no. 2, 524–547. MR 1951054, DOI 10.1137/S1064827501388339
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
Richard E. Ewing, Junping Wang, and Yongjun Yang, A stabilized discontinuous finite element method for elliptic problems, Numer. Linear Algebra Appl. 10 (2003), no. 1-2, 83–104. Dedicated to the 60th birthday of Raytcho Lazarov. MR 1964287, DOI 10.1002/nla.313
Xiaobing Feng and Ohannes A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J. Numer. Anal. 39 (2001), no. 4, 1343–1365. MR 1870847, DOI 10.1137/S0036142900378480
Martin J. Gander, Optimized Schwarz methods, SIAM J. Numer. Anal. 44 (2006), no. 2, 699–731. MR 2218966, DOI 10.1137/S0036142903425409
M. J. Gander and S. Hajian, Block Jacobi for discontinuous Galerkin discretizations: no ordinary Schwarz methods, Domain Decomposition Methods in Science and Engineering XXI, Lect. Notes Comput. Sci. Eng., Springer, 2013.
Martin J. Gander and Soheil Hajian, Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization, SIAM J. Numer. Anal. 53 (2015), no. 1, 573–597. MR 3313831, DOI 10.1137/140961857
Martin J. Gander and Felix Kwok, Best Robin parameters for optimized Schwarz methods at cross points, SIAM J. Sci. Comput. 34 (2012), no. 4, A1849–A1879. MR 2970388, DOI 10.1137/110837218
M. J. Gander and F. Kwok, On the applicability of Lionsâ energy estimates in the analysis of discrete optimized Schwarz methods with cross points, Domain Decomposition Methods in Science and Engineering XX, Springer, 2013, pp. 475–483.
M. J. Gander and K. Santugini, Cross-points in domain decomposition methods with a finite element discretization, revised (2015).
Claude J. Gittelson, Ralf Hiptmair, and Ilaria Perugia, Plane wave discontinuous Galerkin methods: analysis of the $h$-version, M2AN Math. Model. Numer. Anal. 43 (2009), no. 2, 297–331. MR 2512498, DOI 10.1051/m2an/2009002
S. Hajian, An optimized Schwarz algorithm for discontinuous Galerkin methods, Domain Decomposition Methods in Science and Engineering XXII, Springer, 2014.
S. Hajian, Analysis of Schwarz methods for discontinuous Galerkin discretizations, Ph.D. thesis, 06/04 2015, ID: unige:75225.
C. Lehrenfeld, Hybrid discontinuous Galerkin methods for incompressible flow problems, Master’s thesis, RWTH Aachen, 2010.
P.-L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 202–223. MR 1064345
Sébastien Loisel, Condition number estimates for the nonoverlapping optimized Schwarz method and the 2-Lagrange multiplier method for general domains and cross points, SIAM J. Numer. Anal. 51 (2013), no. 6, 3062–3083. MR 3129755, DOI 10.1137/100803316
LiZhen Qin, ZhongCi Shi, and XueJun Xu, On the convergence rate of a parallel nonoverlapping domain decomposition method, Sci. China Ser. A 51 (2008), no. 8, 1461–1478. MR 2426076, DOI 10.1007/s11425-008-0103-2
Lizhen Qin and Xuejun Xu, On a parallel Robin-type nonoverlapping domain decomposition method, SIAM J. Numer. Anal. 44 (2006), no. 6, 2539–2558. MR 2272605, DOI 10.1137/05063790X
Lizhen Qin and Xuejun Xu, Optimized Schwarz methods with Robin transmission conditions for parabolic problems, SIAM J. Sci. Comput. 31 (2008), no. 1, 608–623. MR 2460791, DOI 10.1137/070682149
Andrea Toselli and Olof Widlund, Domain decomposition methods—algorithms and theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. MR 2104179, DOI 10.1007/b137868