Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems : non-overlapping case
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, p. 21-54

We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h), where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345-357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.

DOI : https://doi.org/10.1051/m2an:2007006
Classification:  65N30,  65N55
Keywords: domain decomposition methods, discontinuous Galerkin, elliptic problems
@article{M2AN_2007__41_1_21_0,
     author = {Antonietti, Paola F. and Ayuso, Blanca},
     title = {Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems : non-overlapping case},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {1},
     year = {2007},
     pages = {21-54},
     doi = {10.1051/m2an:2007006},
     zbl = {1129.65080},
     mrnumber = {2323689},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2007__41_1_21_0}
}
Antonietti, Paola F.; Ayuso, Blanca. Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems : non-overlapping case. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, pp. 21-54. doi : 10.1051/m2an:2007006. http://www.numdam.org/item/M2AN_2007__41_1_21_0/

[1] R.A. Adams, Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, Pure and Applied Mathematics, Vol. 65 (1975). | MR 450957 | Zbl 0314.46030

[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. | Zbl pre05194187

[3] D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742-760. | Zbl 0482.65060

[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 (2001/02) 1749-1779 (electronic). | Zbl 1008.65080

[5] I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863-875. | Zbl 0237.65066

[6] 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. | Zbl 0871.76040

[7] 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., Antwerpen, Belgium (1997) 99-108, Technologisch Instituut.

[8] 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. | Zbl 0924.76051

[9] S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41 (2003) 306-324 (electronic). | Zbl 1045.65100

[10] S.C. Brenner and K. Wang, Two-level additive Schwarz preconditioners for C 0 interior penalty methods. Numer. Math. 102 (2005) 231-255. | Zbl 1088.65108

[11] F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16 (2000) 365-378. | Zbl 0957.65099

[12] F. Brezzi, L.D. Marini and E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Methods Appl. Sci. 14 (2004) 1893-1903. | Zbl 1070.65117

[13] X.-C. Cai and O.B. Widlund, Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Statist. Comput. 13 (1992) 243-258. | Zbl 0746.65085

[14] P.E. Castillo, Local Discontinuous Galerkin methods for convection-diffusion and elliptic problems. Ph.D. thesis, University of Minnesota, Minneapolis (2001).

[15] P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676-1706 (electronic). | Zbl 0987.65111

[16] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, Studies in Mathematics and its Applications, Vol. 4 (1978). | MR 520174 | Zbl 0383.65058

[17] B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, in High-order methods for computational physics, Springer, Berlin, Lect. Notes Comput. Sci. Eng. 9 (1999) 69-224. | Zbl 0937.76049

[18] 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). | Zbl 0927.65118

[19] B. Cockburn, G.E. Karniadakis, and C.-W. Shu. The development of discontinuous Galerkin methods, in Discontinuous Galerkin methods (Newport, RI, 1999), Springer, Berlin, Lect. Notes Comput. Sci. Eng. 11 (2000) 3-50. | Zbl 0989.76045

[20] C. Dawson, S. Sun and M.F. Wheeler, Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565-2580. | Zbl 1067.76565

[21] 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), Springer, Berlin, Lect. Notes Phys. 58 (1976) 207-216.

[22] 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. | Zbl 0524.65019

[23] 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). | Zbl 1007.65104

[24] X. Feng and O.A. Karakashian, Analysis of two-level overlapping additive Schwarz preconditioners for a discontinuous Galerkin method. In Domain decomposition methods in science and engineering (Lyon, 2000), Theory Eng. Appl. Comput. Methods, Internat. Center Numer. Methods Eng. (CIMNE), Barcelona (2002) 237-245. | Zbl 1026.65121

[25] G.H. Golub and C.F. Van Loan, Matrix computations. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, third edition (1996). | MR 1417720 | Zbl 0865.65009

[26] J. Gopalakrishnan and G. Kanschat. Application of unified DG analysis to preconditioning DG methods, in Computational Fluid and Solid Mechanics 2003, K.J. Bathe Ed., Elsevier (2003) 1943-1945.

[27] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math. 95 (2003) 527-550. | Zbl 1044.65084

[28] B. Heinrich and K. Pietsch, Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68 (2002) 217-238. | Zbl 1002.65124

[29] P. Houston and E. Süli, hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. SIAM J. Sci. Comput. 23 (2001) 1226-1252 (electronic). | Zbl 1029.65130

[30] 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). | Zbl 1038.65135

[31] P. Le Tallec, Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1 (1994) 121-220. | Zbl 0802.73079

[32] 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. | Zbl 0658.65090

[33] P.-L. Lions, On the Schwarz alternating method. II. Stochastic interpretation and order properties, in Domain decomposition methods (Los Angeles, CA, 1988), SIAM, Philadelphia, PA (1989) 47-70. | Zbl 0681.65072

[34] P.-L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), SIAM, Philadelphia, PA (1990) 202-223. | Zbl 0704.65090

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

[36] 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. | Zbl 0951.65108

[37] B. Rivière, M.F. Wheeler and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 902-931 (electronic). | Zbl 1010.65045

[38] Y. Saad and M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7 (1986) 856-869. | Zbl 0599.65018

[39] M. Sarkis and D.B. Szyld, Optimal left and right additive Schwarz preconditioning for Minimal Residual methods with euclidean and energy norms. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1612-1621.

[40] B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain decomposition. Cambridge University Press, Cambridge, Parallel multilevel methods for elliptic partial differential equations (1996). | MR 1410757 | Zbl 0857.65126

[41] G. Starke, Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems. Numer. Math. 78 (1997) 103-117. | Zbl 0888.65037

[42] R. Stenberg, Mortaring by a method of J. A. Nitsche, in Computational mechanics (Buenos Aires, 1998), pages CD-ROM file. Centro Internac. Métodos Numér. Ing., Barcelona (1998).

[43] A. Toselli and O. Widlund, Domain decomposition methods-algorithms and theory, Springer Series in Computational Mathematics 34, Springer-Verlag, Berlin (2005). | Zbl 1069.65138

[44] M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152-161. | Zbl 0384.65058

[45] J.H. Wilkinson, The algebraic eigenvalue problem. Monographs on Numerical Analysis, The Clarendon Press Oxford University Press, New York (1988), Oxford Science Publications. | MR 950175 | Zbl 0626.65029

[46] J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581-613. | Zbl 0788.65037

[47] 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). | Zbl 0999.47015

[48] J. Xu and J. Zou. Some nonoverlapping domain decomposition methods. SIAM Rev. 40 (1998) 857-914 (electronic). | Zbl 0913.65115