A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1557-1581.

In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt's class A2. The theory hinges on local approximation properties of either Clément or Scott-Zhang interpolation operators, without need of modifications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good effectivity indices.

DOI : 10.1051/m2an/2014010
Classification : 35J15, 65N12, 65N15, 65N30, 65N50, 65Y20
Mots-clés : elliptic problems, point sources, a posteriori error estimates, finite elements, weighted Sobolev spaces
@article{M2AN_2014__48_6_1557_0,
     author = {Agnelli, Juan Pablo and Garau, Eduardo M. and Morin, Pedro},
     title = {\protect\emph{A posteriori }error estimates for elliptic problems with {Dirac} measure terms in weighted spaces},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1557--1581},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {6},
     year = {2014},
     doi = {10.1051/m2an/2014010},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2014010/}
}
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Agnelli, Juan Pablo; Garau, Eduardo M.; Morin, Pedro. A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1557-1581. doi : 10.1051/m2an/2014010. http://archive.numdam.org/articles/10.1051/m2an/2014010/

[1] Th. 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 | Zbl

[2] Th. Apel, A.-M. Sändig, J.R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19 (1996) 63-85. | MR | Zbl

[3] R. Araya, E. Behrens, R. Rodríguez, An adaptive stabilized finite element scheme for a water quality model. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2800-2812. | MR | Zbl

[4] R. Araya, E. Behrens, R. Rodríguez, A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math. 105 (2006) 193-216. | MR | Zbl

[5] I. Babuška, Error-Bounds for Finite Element Method. Numer. Math. 16 (1971) 322-333. | MR | Zbl

[6] I. Babuška, M.B. Rosenzweig, A finite element scheme for domains with corners. Numer. Math. 20 (1972/73) 1-21. | MR | Zbl

[7] Z. Belhachmi, C. Bernardi, S. Deparis, Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Numer. Math. 105 (2006) 217-247. | MR | Zbl

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

[9] P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique 9 (1975) 77-84. | Numdam | MR | Zbl

[10] C. D'Angelo, Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: Applications to one- and three-dimensional coupled problems. SIAM J. Numer. Anal. 50 (2012) 194-215. | MR | Zbl

[11] C. D'Angelo, A. Quarteroni, On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18 (2008) 1481-1504. | MR

[12] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl

[13] K. Eriksson, Improved accuracy by adapted mesh-refinements in the finite element method. Math. Comput. 44 (1985) 321-343. | MR | Zbl

[14] L.C. Evans, Partial diferential equations. Grad. Stud. Math., vol. 19. American Mathematical Society, Providence, RI (1998). | MR | Zbl

[15] E. Fabes, C. Kenig, R. Serapioni, The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equations 7 (1982) 77-116. | MR | Zbl

[16] F. Gaspoz, P. Morin, A. Veeser, A posteriori error estimates with point sources, in preparation (2013).

[17] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer-Verlag, Berlin (1983). | MR | Zbl

[18] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations. Oxford Science Publications (1993). | MR | Zbl

[19] T. Kilpeläinen, Smooth approximation in weighted Sobolev spaces, Comment. Math. Univ. Carolinae 38 (1997) 29-35. | Zbl

[20] V.A. Kozlov, V.G. Maz'Ya, J. Rossmann, Elliptic boundary value problems in domains with point singularities, Math. Surv. Monogr., vol. 52. American Mathematical Society, Providence, RI (1997). | MR | Zbl

[21] A. Kufner, Weighted Sobolev spaces, A Wiley-Interscience Publication. John Wiley & Sons Inc., New York (1985). Translated from the Czech. | MR | Zbl

[22] P. Morin, K.G. Siebert, A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707-737. | MR | Zbl

[23] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972) 207-226. | MR | Zbl

[24] B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192 (1974) 261-274. | MR | Zbl

[25] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. Ser. 16 (1962) 305-326. | Numdam | MR | Zbl

[26] R.H. Nochetto, K.G. Siebert, A. Veeser, Theory of adaptive finite element methods: an introduction. Edited by R.A. DeVore, A. Kunoth. Multiscale, nonlinear and adaptive approximation. Springer, Berlin (2009) 409-542. | MR | Zbl

[27] B. Opic, A. Kufner, Hardy-type inequalities, Pitman Res. Notes Math. Ser., vol. 219. Longman Scientific & Technical, Harlow (1990). | MR | Zbl

[28] L.R. Scott, Finite Element Convergence for Singular Data. Numer. Math. 21 (1973) 317-327. | MR | Zbl

[29] K.G. Siebert, A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal. 31 (2011) 947-970. | MR | Zbl

[30] L.R. Scott, S. Zhang, Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary Conditions. Math. Comput. 54 (1990) 483-493. | MR | Zbl

[31] T.I. Seidman, M.K. Gobbert, D.W. Trott, M. Kružík, Finite element approximation for time-dependent diffusion with measure-valued source. Numer. Math. 122 (2012) 709-723. | MR | Zbl

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