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A posteriori error estimates for elliptic problems 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.
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Abstract

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.

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

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