Discontinuous Galerkin methods for problems with Dirac delta source
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, p. 1467-1483

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.

DOI : https://doi.org/10.1051/m2an/2012010
Classification:  65N30
Keywords: elliptic pdes, discontinuous Galerkin methods, Dirac delta source
@article{M2AN_2012__46_6_1467_0,
     author = {Houston, Paul and Wihler, Thomas Pascal},
     title = {Discontinuous Galerkin methods for problems with Dirac delta source},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {6},
     year = {2012},
     pages = {1467-1483},
     doi = {10.1051/m2an/2012010},
     zbl = {1272.65092},
     mrnumber = {2996336},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_6_1467_0}
}
Houston, Paul; Wihler, Thomas Pascal. Discontinuous Galerkin methods for problems with Dirac delta source. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, pp. 1467-1483. doi : 10.1051/m2an/2012010. http://www.numdam.org/item/M2AN_2012__46_6_1467_0/

[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier (2003). | MR 2424078 | Zbl 1098.46001

[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. | MR 2812554 | Zbl 1229.65203

[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. | MR 2262756 | Zbl 1162.65401

[4] D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742-760. | MR 664882 | Zbl 0482.65060

[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. | MR 1885715 | Zbl 1008.65080

[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. | MR 2009692 | Zbl 1105.65349

[7] E. Casas, L2 estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47 (1985) 627-632. | MR 812624 | Zbl 0561.65071

[8] M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lect. Notes Math. 1341 (1988). | MR 961439 | Zbl 0668.35001

[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. | MR 440955

[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. | MR 1352472 | Zbl 0829.65122

[11] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). | MR 775683 | Zbl 0695.35060

[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). | Zbl 1141.76428

[13] P. Houston and T.P. Wihler, Second-order elliptic PDE with discontinuous boundary data. IMA J. Numer. Anal. 32 (2012) 48-74. | MR 2875243 | Zbl 1237.65124

[14] V. John, A posteriori L2-error estimates for the nonconforming P1/P0-finite element discretization of the Stokes equations. J. Comput. Appl. Math. 96 (1998) 99-116. | MR 1650444 | Zbl 0930.65123

[15] B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory and Implementation, in Front. Appl. Math. SIAM (2008). | Zbl 1153.65112

[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). | MR 1860450 | Zbl 1010.65045

[17] R. Scott, Finite element convergence for singular data. Numer. Math. 21 (1973/1974) 317-327. | MR 337032 | Zbl 0255.65037

[18] M.F. Wheeler, An elliptic collocation finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152-161. | MR 471383 | Zbl 0384.65058

[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. | MR 2753240 | Zbl 1228.65227