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.

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] Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: Non-overlapping case. ESAIM: M2AN 41 (2007) 21-54. | EuDML | Numdam | MR | Zbl

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

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

,[4] Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749-1779 (electronic). | MR | Zbl

, , and ,[5] 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

and ,[6] 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.

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

and ,[8] Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp. 57 (1991) 1-21. | MR | Zbl

, , and ,[9] 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

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

and ,[11] Multigrid algorithms for ${C}^{0}$ interior penalty methods. SIAM J. Numer. Anal. 44 (2006) 199-223 (electronic). | MR | Zbl

and ,[12] Two-level additive Schwarz preconditioners for ${C}^{0}$ interior penalty methods. Numer. Math. 102 (2005) 231-255. | MR | Zbl

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

and ,[14] Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations 16 (2000) 365-378. | MR | Zbl

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

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

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

and ,[18] Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565-2580. | MR | Zbl

, and ,[19] Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numer. Linear Algebra Appl. 13 (2006) 753-770. | MR

, , and ,[20] 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

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

, and ,[22] 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

and ,[23] Matrix Computations. 3rd Edn., Johns Hopkins University Press, Baltimore, USA (1996). | MR | Zbl

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

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

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

,[27] 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

and ,[28] 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] 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

and ,[30] 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

and ,[31] 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

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

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

, and ,[34] 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

and ,[35] Domain decomposition. Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (1996). | MR | Zbl

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

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

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

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

,[41] 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

and ,*Cited by Sources: *