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Discontinuous Galerkin methods for problems with Dirac deltasource∗

Published online by Cambridge University Press:  31 May 2012

Paul Houston  and
Thomas Pascal Wihler
Paul Houston
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. [email protected]
Thomas Pascal Wihler
Affiliation:
Mathematics Institute, University of Bern, 3012 Bern, Switzerland; [email protected]
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Abstract

In this article we study discontinuous Galerkin finite element discretizations of linearsecond-order elliptic partial differential equations with Dirac delta right-hand side. Inparticular, assuming that the underlying computational mesh is quasi-uniform, we derive ana priori bound on the error measured in terms of theL2-norm. Additionally, we develop residual-based aposteriori error estimators that can be used within an adaptive mesh refinementframework. Numerical examples for the symmetric interior penalty scheme are presentedwhich confirm the theoretical results.

Keywords

Elliptic PDEsdiscontinuous Galerkin methodsDirac delta source
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1467 - 1483
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier (2003).
Apel, T., Benedix, O., Sirch, D. and Vexler, B., A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 49 (2011) 992–1005. Google Scholar
Araya, R., E. Behrens and R. Rodríguez. A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math. 105 (2006) 193–216. Google Scholar
Arnold, D.N., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. Google Scholar
Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749–1779. Google Scholar
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.
Casas, E., L 2 estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47 (1985) 627–632. Google Scholar
Dauge, M., Elliptic Boundary Value Problems on Corner Domains. Lect. Notes Math. 1341 (1988). Google Scholar
Douglas, J. and Dupont, T., 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. Google Scholar
Eriksson, K., Estep, D., Hansbo, P. and Johnson, C., Introduction to adaptive methods for differential equations, edited by A. Iserles. Cambridge University Press. Acta Numerica (1995) 105–158. Google Scholar
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985).
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).
Houston, P. and Wihler, T.P., Second-order elliptic PDE with discontinuous boundary data. IMA J. Numer. Anal. 32 (2012) 48–74. Google Scholar
John, V., A posteriori L 2-error estimates for the nonconforming P 1/P 0-finite element discretization of the Stokes equations. J. Comput. Appl. Math. 96 (1998) 99–116. Google Scholar
B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory and Implementation, in Front. Appl. Math. SIAM (2008).
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) 902–931 (electronic). Google Scholar
Scott, R., Finite element convergence for singular data. Numer. Math. 21 (1973/1974) 317–327. Google Scholar
Wheeler, M.F., An elliptic collocation finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. Google Scholar
Wihler, T.P. and Rivière, B., Discontinuous Galerkin methods for second-order elliptic PDE with low-regularity solutions. J. Sci. Comput. 46 (2011) 151–165. Google Scholar