Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T14:54:07.655Z Has data issue: false hasContentIssue false
  • English
  • Français

A posteriori error estimates for ellipticproblems with Dirac measure terms in weighted spaces

Published online by Cambridge University Press:  09 September 2014

Juan Pablo Agnelli ,
Eduardo M. Garau  and
Pedro Morin
Juan Pablo Agnelli
Affiliation:
Instituto de Matemática Aplicada del Litoral, Universidad Nacional del Litoral and CONICET. IMAL, Colectora Ruta Nac. N. 168, Paraje El Pozo, 3000 Santa Fe, Argentina. . jpagnelli@santafe-conicet.gov.ar,egarau@santafe-conicet.gov.ar,pmorin@santafe-conicet.gov.ar Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Argentina.
Eduardo M. Garau
Affiliation:
Instituto de Matemática Aplicada del Litoral, Universidad Nacional del Litoral and CONICET. IMAL, Colectora Ruta Nac. N. 168, Paraje El Pozo, 3000 Santa Fe, Argentina. . jpagnelli@santafe-conicet.gov.ar,egarau@santafe-conicet.gov.ar,pmorin@santafe-conicet.gov.ar Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, Argentina.
Pedro Morin
Affiliation:
Instituto de Matemática Aplicada del Litoral, Universidad Nacional del Litoral and CONICET. IMAL, Colectora Ruta Nac. N. 168, Paraje El Pozo, 3000 Santa Fe, Argentina. . jpagnelli@santafe-conicet.gov.ar,egarau@santafe-conicet.gov.ar,pmorin@santafe-conicet.gov.ar Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, Argentina.
Get access

Abstract

In this article we develop a posteriori error estimates for second orderlinear elliptic problems with point sources in two- and three-dimensional domains. Weprove a global upper bound and a local lower bound for the error measured in a weightedSobolev space. The weight considered is a (positive) power of the distance to the supportof the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theoryhinges on local approximation properties of either Clément or Scott–Zhang interpolationoperators, without need of modifications, and makes use of weighted estimates forfractional integrals and maximal functions. Numerical experiments with an adaptivealgorithm yield optimal meshes and very good effectivity indices.

Keywords

Elliptic problemspoint sourcesa posteriori error estimatesfinite elementsweighted Sobolev spaces
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1557 - 1581
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Apel, Th., 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
Apel, Th., Sändig, A.-M., Whiteman, J.R., 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. Google Scholar
Araya, R., Behrens, E., Rodríguez, R., An adaptive stabilized finite element scheme for a water quality model. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2800–2812. Google Scholar
Araya, R., Behrens, E., Rodríguez, R., A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math. 105 (2006) 193–216. Google Scholar
Babuška, I., Error-Bounds for Finite Element Method. Numer. Math. 16 (1971) 322–333. Google Scholar
Babuška, I., Rosenzweig, M.B., A finite element scheme for domains with corners. Numer. Math. 20 (1972/73) 1–21. Google Scholar
Belhachmi, Z., Bernardi, C., Deparis, S., Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Numer. Math. 105 (2006) 217–247. Google Scholar
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
Clément, P., Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique 9 (1975) 77–84. Google Scholar
D’Angelo, C., 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. Google Scholar
D’Angelo, C., Quarteroni, A., On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18 (2008) 1481–1504. Google Scholar
Dörfler, W., A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. Google Scholar
Eriksson, K., Improved accuracy by adapted mesh-refinements in the finite element method. Math. Comput. 44 (1985) 321–343. Google Scholar
L.C. Evans, Partial diferential equations. Grad. Stud. Math., vol. 19. American Mathematical Society, Providence, RI (1998).
Fabes, E., Kenig, C., Serapioni, R., The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equations 7 (1982) 77–116. Google Scholar
F. Gaspoz, P. Morin, A. Veeser, A posteriori error estimates with point sources, in preparation (2013).
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).
J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations. Oxford Science Publications (1993).
Kilpeläinen, T., Smooth approximation in weighted Sobolev spaces, Comment. Math. Univ. Carolinae 38 (1997) 29–35. Google Scholar
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).
A. Kufner, Weighted Sobolev spaces, A Wiley-Interscience Publication. John Wiley & Sons Inc., New York (1985). Translated from the Czech.
Morin, P., Siebert, K.G., Veeser, A., A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707–737. Google Scholar
Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972) 207–226. Google Scholar
Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192 (1974) 261–274. Google Scholar
Nečas, J., 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. Google Scholar
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.
B. Opic, A. Kufner, Hardy-type inequalities, Pitman Res. Notes Math. Ser., vol. 219. Longman Scientific & Technical, Harlow (1990).
Scott, L.R., Finite Element Convergence for Singular Data. Numer. Math. 21 (1973) 317–327. Google Scholar
Siebert, K.G., A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal. 31 (2011) 947–970. Google Scholar
Scott, L.R., Zhang, S., Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary Conditions. Math. Comput. 54 (1990) 483–493. Google Scholar
Seidman, T.I., Gobbert, M.K., Trott, D.W., Kružík, M., Finite element approximation for time-dependent diffusion with measure-valued source. Numer. Math. 122 (2012) 709–723. Google Scholar