Discontinuous Galerkin methods for problems with Dirac delta source∗

Paul Houston; Thomas Pascal Wihler

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 6, page 1467-1483
  • ISSN: 0764-583X

Abstract

top
In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.

How to cite

top

Houston, Paul, and Wihler, Thomas Pascal. "Discontinuous Galerkin methods for problems with Dirac delta source∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1467-1483. <http://eudml.org/doc/276387>.

@article{Houston2012,
abstract = {In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.},
author = {Houston, Paul, Wihler, Thomas Pascal},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Elliptic PDEs; discontinuous Galerkin methods; Dirac delta source; discontinous Galerkin methods; a priori error bound; numerical results; finite element; linear second-order elliptic PDE},
language = {eng},
month = {5},
number = {6},
pages = {1467-1483},
publisher = {EDP Sciences},
title = {Discontinuous Galerkin methods for problems with Dirac delta source∗},
url = {http://eudml.org/doc/276387},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Houston, Paul
AU - Wihler, Thomas Pascal
TI - Discontinuous Galerkin methods for problems with Dirac delta source∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/5//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1467
EP - 1483
AB - In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.
LA - eng
KW - Elliptic PDEs; discontinuous Galerkin methods; Dirac delta source; discontinous Galerkin methods; a priori error bound; numerical results; finite element; linear second-order elliptic PDE
UR - http://eudml.org/doc/276387
ER -

References

top
  1. R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier (2003).  
  2. T. Apel, O. Benedix, D. Sirch and B. Vexler, A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal.49 (2011) 992–1005.  Zbl1229.65203
  3. R. Araya, E. Behrens and R. Rodríguez. A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math.105 (2006) 193–216.  Zbl1162.65401
  4. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760.  Zbl0482.65060
  5. 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) 1749–1779.  Zbl1008.65080
  6. R. Becker and R. Rannacher, An optimal control approach to a-posteriori error estimation in finite element methods, edited by A. Iserles. Cambridge University Press. Acta Numerica (2001) 1–102.  Zbl1105.65349
  7. E. Casas, L2 estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math.47 (1985) 627–632.  Zbl0561.65071
  8. M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lect. Notes Math.1341 (1988).  Zbl0668.35001
  9. J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975). Lect. Notes Phys.58 (1976) 207–216.  
  10. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, edited by A. Iserles. Cambridge University Press. Acta Numerica (1995) 105–158.  Zbl0829.65122
  11. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985).  Zbl0695.35060
  12. P. Houston and E. Süli, Adaptive finite element approximation of hyperbolic problems, in Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, edited by T. Barth and H. Deconinck. Lect. Notes Comput. Sci. Eng.25 (2002).  
  13. P. Houston and T.P. Wihler, Second-order elliptic PDE with discontinuous boundary data. IMA J. Numer. Anal.32 (2012) 48–74.  Zbl1237.65124
  14. V. John, A posterioriL2-error estimates for the nonconforming P1/P0-finite element discretization of the Stokes equations. J. Comput. Appl. Math.96 (1998) 99–116.  
  15. B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory and Implementation, in Front. Appl. Math. SIAM (2008).  Zbl1153.65112
  16. 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).  Zbl1010.65045
  17. R. Scott, Finite element convergence for singular data. Numer. Math.21 (1973/1974) 317–327.  Zbl0255.65037
  18. M.F. Wheeler, An elliptic collocation finite element method with interior penalties. SIAM J. Numer. Anal.15 (1978) 152–161.  Zbl0384.65058
  19. T.P. Wihler and B. Rivière, Discontinuous Galerkin methods for second-order elliptic PDE with low-regularity solutions. J. Sci. Comput.46 (2011) 151–165.  Zbl1228.65227

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.