Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A new error analysis for discontinuous finite element methods for linear elliptic problems

Author: Thirupathi Gudi
Journal: Math. Comp. 79 (2010), 2169-2189
MSC (2010): Primary 65N30, 65N15
Published electronically: April 12, 2010
MathSciNet review: 2684360
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only the $ H^k$ weak formulation of a boundary value problem of order $ 2k$. This is accomplished by replacing the Galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in $ H^k$.

References [Enhancements On Off] (What's this?)

  • 1. M. Ainsworth and J. T. Oden.
    A posteriori error estimation in finite element analysis.
    Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308 (2003b:65001)
  • 2. D.N. Arnold.
    An interior penalty finite element method with discontinuous elements.
    SIAM J. Numer. Anal., 19:742-760, 1982. MR 664882 (83f:65173)
  • 3. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini.
    Unified analysis of discontinuous Galerkin methods for elliptic problems.
    SIAM J. Numer. Anal., 39:1749-1779, 2002. MR 1885715 (2002k:65183)
  • 4. G.A. Baker.
    Finite element methods for elliptic equations using nonconforming elements.
    Math. Comp., 31:45-59, 1977. MR 0431742 (55:4737)
  • 5. A. Berger, R. Scott and G. Strang.
    Approximate boundary conditions in the finite element method.
    In: Symposia Mathematica, Vol. X (Convegno di Analasi Numerica), London, Academic Press 1972, 295-313, 1972. MR 0403258 (53:7070)
  • 6. S.C. Brenner.
    Two-level additive Schwarz preconditioners for nonconforming finite element methods.
    Math. Comp., 65:897-921, 1996. MR 1348039 (96j:65117)
  • 7. S.C. Brenner.
    Convergence of nonconforming multigrid methods without full elliptic regularity.
    Math. Comp., 68:25-53, 1999. MR 1620215 (99c:65229)
  • 8. S.C. Brenner.
    Ponicaré-Friedrichs inequalities for piecewise $ H^1$ functions.
    SIAM J. Numer. Anal, 41:306-324, 2003. MR 1974504 (2004d:65140)
  • 9. S.C. Brenner, K. Wang and J. Zhao.
    Poincaré-Friedrichs inequalities for piecewise $ H^2$ functions.
    Numer. Funct. Anal. Optim., 25: 463-478, 2004. MR 2106270 (2005i:65178)
  • 10. S.C. Brenner.
    Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions.
    Elec. Trans. Numer. Anal., 18: 42-48, 2004. MR 2083293 (2005k:65239)
  • 11. S.C. Brenner and L.R. Scott.
    The Mathematical Theory of Finite Element Methods $ ($Third Edition$ )$.
    Springer-Verlag, New York, 2008. MR 2373954 (2008m:65001)
  • 12. S.C. Brenner and L.-Y. Sung.
    $ C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains.
    J. Sci. Comput., 22/23:83-118, 2005. MR 2142191 (2005m:65258)
  • 13. S.C. Brenner and L. Owens.
    A weakly over-penalized non-symmetric interior penalty method.
    J. Numer. Anal. Indust. Appl. Math., 2: 35-48, 2007. MR 2332345 (2008c:65315)
  • 14. S.C. Brenner and L. Owens.
    A $ W$-cycle algorithm for weakly over-penalized interior penalty method.
    Comput. Meth. Appl. Mech. engrg., 196: 3823-3832, 2007. MR 2340007 (2008i:65286)
  • 15. S.C. Brenner L. Owens and L.-Y. Sung.
    A weakly over-penalized symmetric interior penalty method.
    E. Tran. Numer. Anal, 30: 107-127, 2008. MR 2480072 (2009k:65236)
  • 16. S.C. Brenner, T. Gudi and L.-Y. Sung.
    An a posteriori error estimator for a quadratic $ C^0$ interior penalty method for the biharmonic problem.
    to appear in IMA J. Numer. Anal., doi:10.1093/imanum/drn057.
  • 17. 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. , 38:1676-1706, 2000. MR 1813251 (2002k:65175)
  • 18. C. Carstensen, T. Gudi and M. Jensen.
    A unifying theory of a posteriori error control for discontinuous Galerkin FEMs.
    Numer. Math., 112:363-379, 2009. MR 2501309
  • 19. P.G. Ciarlet.
    The Finite Element Method for Elliptic Problems.
    North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • 20. B. Cockburn and C.W. Shu.
    The local discontinuous Galerkin method for time-dependent convection-diffusion systems.
    SIAM J. Numer. Anal., 35:2440-2463, 1998. MR 1655854 (99j:65163)
  • 21. M. Crouzeix and P.A. Raviart.
    Conforming and Nonconforming finite element methods for solving the stationary Stokes equations.
    RAIRO 7, rev. 3:33-76, 1973. MR 0343661 (49:8401)
  • 22. J. Douglas, Jr. and T. Dupont.
    Interior penalty procedures for elliptic and parabolic Galerkin methods.
    Lecture Notes in Phys., 58, Springer-Verlag, Berlin, 1976. MR 0440955 (55:13823)
  • 23. G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei, and R.L. Taylor.
    Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity.
    Comput. Methods Appl. Mech. Engrg., 191:3669-3750, 2002. MR 1915664 (2003d:74086)
  • 24. X. Feng and O.A. Karakashian.
    Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation in phase seperation.
    Math. Comp., 76:1093-1117, 2007. MR 2299767 (2008a:74048)
  • 25. X. Feng and O.A. Karakashian.
    Two-level nonoverlapping additive Schwarz methods for a discontinuous Galerkin approximation of the biharmonic problem.
    J. Sci. Comp., 22:299-324, 2005. MR 2142199 (2006b:65156)
  • 26. E. H. Georgoulis P. Houston, and J. Virtanen.
    An a posteriori error indicator for discontinuous Galerkin approximations of fourth order elliptic problems.
    Nottingham eprint.
  • 27. P. Grisvard.
    Elliptic Problems in Nonsmooth Domains.
    Pitman, Boston, 1985. MR 775683 (86m:35044)
  • 28. T. Gudi, N. Nataraj and A. K. Pani.
    Mixed discontinuous Galerkin methods for the biharmonic equation.
    J. Sci. Comput., 37:103-232, 2008. MR 2453216 (2009j:65319)
  • 29. P. Houston, D. Schötzau, and T.P. Wihler.
    Energy norm a posteriori error estimation of $ hp$-adaptive discontinuous Galerkin methods for elliptic problems.
    Math. Models Methods Appl. Sci., pages 33-62, 2007. MR 2290408 (2008a:65216)
  • 30. O.A. Karakashian and F. Pascal.
    A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems.
    SIAM J. Numer. Anal., 41:2374-2399 (electronic), 2003. MR 2034620 (2005d:65192)
  • 31. R.B. Kellogg.
    Singularities in interface problems, in: B. Hubbard (Ed.), Numerical Solutions of Partial Differential Equations II,
    Academic Press, New York, pp. 351-400, 1971. MR 0289923 (44:7108)
  • 32. L.S.D. Morley.
    The triangular equilibrium problem in the solution of plate bending problems.
    Aero. Quart., 19:149-169, 1968.
  • 33. I. Mozolevski, E. Süli, and P.R. Bösing.
    $ hp$-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation.
    J. Sci. Comput., 30:465-491, 2007. MR 2295480 (2008c:65341)
  • 34. L. Owens.
    Multigrid Methods for Weakly Over-Penalized Interior penalty Methods.
    Ph.D. thesis, Department of Mathematics, University of South Corolina, 2007.
  • 35. S. Prudhomme, F. Pascal and J.T. Oden.
    Review of error estimation for discontinuous Galerkin methods.
    TICAM Report 00-27, October 17, 2000.
  • 36. 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., 39:902-931, 2001. MR 1860450 (2002g:65149)
  • 37. R. Verfürth.
    A posteriori error estimation and adaptive mesh-refinement techniques.
    In Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), volume 50, pages 67-83, 1994. MR 1284252 (95c:65171)
  • 38. R. Verfürth.
    A Review of A Posteriori Error Estmation and Adaptive Mesh-Refinement Techniques.
    Wiley-Teubner, Chichester, 1995.
  • 39. M.F. Wheeler.
    An elliptic collocation-finite-element method with interior penalties.
    SIAM J. Numer. Anal., 15:152-161, 1978. MR 0471383 (57:11117)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N15

Retrieve articles in all journals with MSC (2010): 65N30, 65N15

Additional Information

Thirupathi Gudi
Affiliation: Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803

Keywords: Optimal error estimates, elliptic regularity, finite element, discontinuous Galerkin, nonconforming
Received by editor(s): January 5, 2009
Received by editor(s) in revised form: June 16, 2009
Published electronically: April 12, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society