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Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

  • Paola F. Antonietti (a1) and Blanca Ayuso (a2)

Abstract


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.


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[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).
[2] Antonietti, P.F., Buffa, A. and Perugia, I., Discontinuous Galerkin approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Engrg. 195 (2006) 34833503.
[3] Arnold, D.N., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742760.
[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).
[5] Babuška, I. and Zlámal, M., Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863875.
[6] Bassi, F. and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267279.
[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] Baumann, C.E. and Oden, J.T., A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311341.
[9] Brenner, S.C., Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41 (2003) 306324 (electronic).
[10] Brenner, S.C. and Wang, K., Two-level additive Schwarz preconditioners for C 0 interior penalty methods. Numer. Math. 102 (2005) 231255.
[11] Brezzi, F., Manzini, G., Marini, D., Pietra, P. and Russo, A., Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16 (2000) 365378.
[12] Brezzi, F., Marini, L.D. and Süli, E., Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Methods Appl. Sci. 14 (2004) 18931903.
[13] Cai, X.-C. and Widlund, O.B., Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Statist. Comput. 13 (1992) 243258.
[14] P.E. Castillo, Local Discontinuous Galerkin methods for convection-diffusion and elliptic problems. Ph.D. thesis, University of Minnesota, Minneapolis (2001).
[15] Castillo, P., Cockburn, B., Perugia, I. and Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 16761706 (electronic).
[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).
[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.
[18] Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 24402463 (electronic).
[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.
[20] Dawson, C., Sun, S. and Wheeler, M.F., Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. 193 (2004) 25652580.
[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] Eisenstat, S.C., Elman, H.C. and Schultz, M.H., Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20 (1983) 345357.
[23] Feng, X. and Karakashian, O.A., Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39 (2001) 13431365 (electronic).
[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.
[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).
[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] Gopalakrishnan, J. and Kanschat, G., A multilevel discontinuous Galerkin method. Numer. Math. 95 (2003) 527550.
[28] Heinrich, B. and Pietsch, K., Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68 (2002) 217238.
[29] Houston, P. and Süli, E., hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. SIAM J. Sci. Comput. 23 (2001) 12261252 (electronic).
[30] Lasser, C. and Toselli, A., An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Math. Comp. 72 (2003) 12151238 (electronic).
[31] Le Tallec, P., Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1 (1994) 121220.
[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.
[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.
[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.
[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] Rivière, B., Wheeler, M.F. and Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3 (1999) 337360.
[37] Rivière, B., Wheeler, M.F. and Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 902931 (electronic).
[38] Saad, Y. and Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7 (1986) 856869.
[39] Sarkis, M. and Szyld, D.B., Optimal left and right additive Schwarz preconditioning for Minimal Residual methods with euclidean and energy norms. Comput. Methods Appl. Mech. Engrg. 196 (2007) 16121621.
[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).
[41] Starke, G., Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems. Numer. Math. 78 (1997) 103117.
[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).
[44] Wheeler, M.F., An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152161.
[45] J.H. Wilkinson, The algebraic eigenvalue problem. Monographs on Numerical Analysis, The Clarendon Press Oxford University Press, New York (1988), Oxford Science Publications.
[46] Iterative, J. Xu methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581613.
[47] Xu, J. and Zikatanov, L., The method of alternating projections and the method of subspace corrections in Hilbert space. J. Amer. Math. Soc. 15 (2002) 573597 (electronic).
[48] J. Xu and J. Zou. Some nonoverlapping domain decomposition methods. SIAM Rev. 40 (1998) 857–914 (electronic).

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Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

  • Paola F. Antonietti (a1) and Blanca Ayuso (a2)

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