Free access
Issue
ESAIM: M2AN
Volume 41, Number 1, January-February 2007
Page(s) 21 - 54
DOI http://dx.doi.org/10.1051/m2an:2007006
Published online 26 April 2007
  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. P.F. Antonietti, A. Buffa and I. Perugia, Discontinuous Galerkin approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3483–3503. [CrossRef] [MathSciNet]
  3. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. [CrossRef] [MathSciNet]
  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. I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863–875. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  9. S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324 (electronic). [CrossRef] [MathSciNet]
  10. S.C. Brenner and K. Wang, Two-level additive Schwarz preconditioners for C0 interior penalty methods. Numer. Math. 102 (2005) 231–255. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  13. X.-C. Cai and O.B. Widlund, Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Statist. Comput. 13 (1992) 243–258. [CrossRef] [MathSciNet]
  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). [CrossRef] [MathSciNet]
  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. 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). [CrossRef] [MathSciNet]
  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. C. Dawson, S. Sun and M.F. Wheeler, Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565–2580. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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). [CrossRef] [MathSciNet]
  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. J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math. 95 (2003) 527–550. [CrossRef] [MathSciNet]
  28. B. Heinrich and K. Pietsch, Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68 (2002) 217–238. [CrossRef] [MathSciNet]
  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). [CrossRef] [MathSciNet]
  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). [CrossRef]
  31. P. Le Tallec, Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1 (1994) 121–220. [MathSciNet]
  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. 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. [NASA ADS] [CrossRef] [EDP Sciences] [MathSciNet] [PubMed]
  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). [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. G. Starke, Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems. Numer. Math. 78 (1997) 103–117. [CrossRef] [MathSciNet]
  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. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. [CrossRef] [MathSciNet]
  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. J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581–613. [CrossRef] [MathSciNet]
  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). [CrossRef]
  48. J. Xu and J. Zou. Some nonoverlapping domain decomposition methods. SIAM Rev. 40 (1998) 857–914 (electronic).

Recommended for you