Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems
ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 3, pp. 443-469.

In this paper we introduce and analyze some non-overlapping multiplicative Schwarz methods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods. For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion on the issue of preconditioning non-symmetric DG approximations of elliptic problems is also included. Extensive numerical experiments to confirm the theoretical results and to assess the robustness and the efficiency of the proposed preconditioners are provided.

DOI: 10.1051/m2an:2008012
Classification: 65N30, 65N55
Keywords: domain decomposition methods, Schwarz preconditioners, discontinuous Galerkin methods
@article{M2AN_2008__42_3_443_0,
     author = {Antonietti, Paola F. and Ayuso, Blanca},
     title = {Multiplicative {Schwarz} methods for discontinuous {Galerkin} approximations of elliptic problems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {443--469},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {3},
     year = {2008},
     doi = {10.1051/m2an:2008012},
     mrnumber = {2423794},
     zbl = {1146.65081},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2008012/}
}
TY  - JOUR
AU  - Antonietti, Paola F.
AU  - Ayuso, Blanca
TI  - Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 443
EP  - 469
VL  - 42
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2008012/
DO  - 10.1051/m2an:2008012
LA  - en
ID  - M2AN_2008__42_3_443_0
ER  - 
%0 Journal Article
%A Antonietti, Paola F.
%A Ayuso, Blanca
%T Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 443-469
%V 42
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2008012/
%R 10.1051/m2an:2008012
%G en
%F M2AN_2008__42_3_443_0
Antonietti, Paola F.; Ayuso, Blanca. Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 3, pp. 443-469. doi : 10.1051/m2an:2008012. http://www.numdam.org/articles/10.1051/m2an:2008012/

[1] P.F. Antonietti and B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: Non-overlapping case. ESAIM: M2AN 41 (2007) 21-54. | EuDML | Numdam | MR | Zbl

[2] P.F. Antonietti, A. Buffa and I. Perugia, Discontinuous Galerkin approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3483-3503. | MR | Zbl

[3] D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742-760. | MR | Zbl

[4] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749-1779 (electronic). | MR | Zbl

[5] F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267-279. | MR | Zbl

[6] F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti and M. Savini, A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, in Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, R. Decuypere and G. Dibelius Eds., Technologisch Instituut, Antwerpen, Belgium (1997) 99-108.

[7] C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311-341. | MR | Zbl

[8] J.H. Bramble, J.E. Pasciak, J.P. Wang and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp. 57 (1991) 1-21. | MR | Zbl

[9] S.C. Brenner and O. Luke, A W-cycle algorithm for a weakly over-penalized interior penalty method. JNAIAM J. Numer. Anal. Indust. Appl. Math 196 (2007) 3823-3832. | MR | Zbl

[10] S.C. Brenner and O. Luke, A weakly over-penalized non-symmetric Interior Penalty method. Comput. Methods Appl. Mech. Engrg. 2 (2007) 35-48. | MR | Zbl

[11] S.C. Brenner and L.-Y. Sung, Multigrid algorithms for C 0 interior penalty methods. SIAM J. Numer. Anal. 44 (2006) 199-223 (electronic). | MR | Zbl

[12] S.C. Brenner and K. Wang, Two-level additive Schwarz preconditioners for C 0 interior penalty methods. Numer. Math. 102 (2005) 231-255. | MR | Zbl

[13] S.C. Brenner and J. Zhao, Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2 (2005) 3-18. | MR | Zbl

[14] F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations 16 (2000) 365-378. | MR | Zbl

[15] X.-C. Cai and O.B. Widlund, Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems. SIAM J. Numer. Anal. 30 (1993) 936-952. | MR | Zbl

[16] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications 4. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl

[17] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440-2463 (electronic). | MR | Zbl

[18] C. Dawson, S. Sun and M.F. Wheeler, Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565-2580. | MR | Zbl

[19] V.A. Dobrev, R.D. Lazarov, P.S. Vassilevski and L.T. Zikatanov, Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numer. Linear Algebra Appl. 13 (2006) 753-770. | MR

[20] J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975), Lecture Notes in Physics 58, Springer, Berlin (1976) 207-216. | MR

[21] S.C. Eisenstat, H.C. Elman and M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20 (1983) 345-357. | MR | Zbl

[22] X. Feng and O.A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1343-1365 (electronic). | MR | Zbl

[23] G.H. Golub and C.F. Van Loan, Matrix Computations. 3rd Edn., Johns Hopkins University Press, Baltimore, USA (1996). | MR | Zbl

[24] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math. 95 (2003) 527-550. | MR | Zbl

[25] G. Kanschat, Preconditioning methods for local discontinuous Galerkin discretizations. SIAM J. Sci. Comput. 25 (2003) 815-831 (electronic). | MR | Zbl

[26] G. Kanschat, Block preconditioners for LDG discretizations of linear incompressible flow problems. J. Sci. Comput. 22/23 (2005) 371-384. | MR | Zbl

[27] C. Lasser and A. Toselli, An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Math. Comp. 72 (2003) 1215-1238 (electronic). | MR | Zbl

[28] P.-L. Lions, On the Schwarz alternating method. I, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987), SIAM, Philadelphia, PA (1988) 1-42. | MR | Zbl

[29] M. Murillo and X.-C. Cai, A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart. Numer. Linear Algebra Appl. 11 (2004) 261-277. | MR | Zbl

[30] L.F. Pavarino and S. Scacchi, Multilevel Schwarz and multigrid preconditioners for the bidomain system, in Domain Decomposition Methods in Science and Engineering XVII, U. Langer, M. Discacciati, D. Keyes, O. Widlund and W. Zulehner Eds., Lecture Notes in Computational Science and Engineering 60, Springer, Heidelberg (2008) 631-638. | MR | Zbl

[31] L.F. Pavarino and A. Toselli, Recent Developments in Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering 23. Springer-Verlag, Berlin (2002). [Selected papers from the Workshop on Domain Decomposition held at ETH Zürich, Zürich, June 7-8 (2001)]. | MR | Zbl

[32] W.H. Reed and T. Hill, Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, USA (1973).

[33] B. Rivière, M.F. Wheeler and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3 (1999) 337-360. | MR | Zbl

[34] V. Simoncini and D.B. Szyld, New conditions for non-stagnation of minimal residual methods. Technical Report 07-04-17, Department of Mathematics, Temple University, USA (2007), to appear in Numerische Mathematik. | MR | Zbl

[35] B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain decomposition. Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (1996). | MR | Zbl

[36] R. Stenberg, Mortaring by a method of J.A. Nitsche, in Computational mechanics (Buenos Aires, 1998), Centro Internac. Métodos Numér. Ing., Barcelona, Spain (1998). | MR

[37] A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer Series in Computational Mathematics 34. Springer-Verlag, Berlin (2005). | MR | Zbl

[38] J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581-613. | MR | Zbl

[39] J. Xu, Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems, in Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991), SIAM, Philadelphia, PA (1992) 106-118. | MR | Zbl

[40] J. Xu, A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM J. Numer. Anal. 29 (1992) 303-319. | MR | Zbl

[41] J. Xu and L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space. J. Amer. Math. Soc. 15 (2002) 573-597 (electronic). | MR | Zbl

Cited by Sources: