Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method Applied to Linear and Nonlinear Diffusion Problems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we present a new residual-based reliable a posteriori error estimator for the local discontinuous Galerkin approximations of linear and nonlinear diffusion problems in polygonal regions of R 2. Our analysis, which applies to convex and nonconvex domains, is based on Helmholtz decompositions of the error and a suitable auxiliary polynomial function interpolating the Dirichlet datum. Several examples confirming the reliability of the estimator and providing numerical evidences for its efficiency are given. Furthermore, the associated adaptive method, which considers meshes with and without hanging nodes, is shown to be much more efficient than a uniform refinement to compute the discrete solutions. In particular, the experiments illustrate the ability of the adaptive algorithm to localize the singularities of each problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • RA Adams (1975) Sobolev Spaces, Pure and Applied Mathematics Academic Press New York–London

    Google Scholar 

  • D. N. Arnold (1982) ArticleTitleInterior penalty finite element method with discontinuous elements SIAM J. Numer. Anal. 19 IssueID4 742–760 Occurrence Handle10.1137/0719052

    Article  Google Scholar 

  • D. N. Arnold F. Brezzi B. Cockburn L. D. Marini (2001) ArticleTitleUnified analysis of discontinuous Galerkin methods for elliptic problems SIAM J. Numer. Anal. 39 IssueID5 1749–1779 Occurrence Handle10.1137/S0036142901384162

    Article  Google Scholar 

  • R. Becker P. Hansbo M. G. Larson (2003) ArticleTitleEnergy norm a posteriori error estimation for discontinuous Galerkin methods Comput. Meth. Appl. Mech. Eng. 192 723–733 Occurrence Handle10.1016/S0045-7825(02)00593-5

    Article  Google Scholar 

  • Becker, R., Hansbo, P., and Stenberg, R. (2001). A finite element method for domain decomposition with non-matching grids. Preprint 2001-15. Chalmers Finite Element Center, Chalmers University of Technology, Sweden.

  • Bustinza, R. (2004). Numerical Analysis of Transmission Problems with Discontinuities (Spanish), Ph.D. thesis, Universidad de Concepción, Concepción, Chile.

  • Bustinza, R., and Gatica, G. N. (2002). A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAM J. Sci. Comput. to appear.

  • C. Carstensen (1997) ArticleTitleAn a posteriori error estimate for a first-kind integral equation Math. Comput. 66 IssueID217 139–155 Occurrence Handle10.1090/S0025-5718-97-00790-4

    Article  Google Scholar 

  • C. Carstensen S. Bartels S. Jansche (2002) ArticleTitleA posteriori error estimates for nonconforming finite element methods Numer. Math. 92 IssueID2 233–256 Occurrence Handle10.1007/s002110100378

    Article  Google Scholar 

  • P. Castillo B. Cockburn I. Perugia D. Schötzau (2000) ArticleTitleAn a priori error analysis of the local discontinuous Galerkin method for elliptic problems SIAM J. Numer. Anal. 38 IssueID5 1676–1706 Occurrence Handle10.1137/S0036142900371003

    Article  Google Scholar 

  • Cockburn, B., and Dawson, C. (2000). Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions. In Whiteman, J. (ed.), Proceedings of the 10th Conference on the Mathematics of Finite Elements and Applications, Elsevier, pp. 225–238.

  • B Cockburn CW Shu (1998) ArticleTitleThe local discontinuous Galerkin finite element method for convection–diffusion systems SIAM J. Numer. Anal 35 IssueID6 2440–2463 Occurrence Handle10.1137/S0036142997316712

    Article  Google Scholar 

  • GN Gatica N Heuer (2001) ArticleTitleAn expanded mixed finite element approach via a dual–dual formulation and the minimum residual method J. Comput. Appl. Math 132 371–385 Occurrence Handle10.1016/S0377-0427(00)00440-4

    Article  Google Scholar 

  • GN Gatica S Meddahi (2001) ArticleTitleA dual–dual mixed formulation for nonlinear exterior transmission problems Math. Comput 70 236

    Google Scholar 

  • Girault, V., and Raviart, P. A., (1986). Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Ser. Comput. Math. 5.

  • J Necas (1986) Introduction to the Theory of Nonlinear Elliptic Equations John Wiley & Sons New York

    Google Scholar 

  • I. Perugia D. Schötzau (2001) ArticleTitleOn the coupling of local discontinuous Galerkin and conforming finite element methods J. Sci. Comput. 16 411–433 Occurrence Handle10.1023/A:1013294207868

    Article  Google Scholar 

  • I. Perugia D. Schötzau (2002) ArticleTitleAn hp-analysis of the local discontinuous Galerkin method for diffusion problems J. Sci. Comput. 17 561–571 Occurrence Handle10.1023/A:1015118613130 Occurrence HandleMR1910752

    Article  MathSciNet  Google Scholar 

  • Riviere, B., and Wheeler, M. F. (2000). A posteriori error estimates and mesh adaptation strategy for discontinuous Galerkin methods applied to diffusion problems, Preprint 00–10. TICAM, University of Texas at Austin, USA.

  • R Verfürth (1996) A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques Wiley–Teubner Chichester, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ingenierí a Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Rommel Bustinza

  2. Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Gabriel N. Gatica

  3. School of Mathematics, University of Minnesota, USA

    Bernardo Cockburn

Authors
  1. Rommel Bustinza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriel N. Gatica
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernardo Cockburn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernardo Cockburn.

Additional information

Mathematics Subject Classifications (1991). 65N30

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bustinza, R., Gatica, G.N. & Cockburn, B. An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method Applied to Linear and Nonlinear Diffusion Problems. J Sci Comput 22, 147–185 (2005). https://doi.org/10.1007/s10915-004-4137-5

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-004-4137-5

Keywords