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

Juan Pablo Agnelli; Eduardo M. Garau; Pedro Morin

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 6, page 1557-1581
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Agnelli, Juan Pablo, Garau, Eduardo M., and Morin, Pedro. "A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.6 (2014): 1557-1581. <http://eudml.org/doc/273164>.

@article{Agnelli2014,
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.},
author = {Agnelli, Juan Pablo, Garau, Eduardo M., Morin, Pedro},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {elliptic problems; point sources; a posteriori error estimates; finite elements; weighted Sobolev spaces; a posteriori error estimates},
language = {eng},
number = {6},
pages = {1557-1581},
publisher = {EDP-Sciences},
title = {A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces},
url = {http://eudml.org/doc/273164},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Agnelli, Juan Pablo
AU - Garau, Eduardo M.
AU - Morin, Pedro
TI - A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 6
SP - 1557
EP - 1581
AB - 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.
LA - eng
KW - elliptic problems; point sources; a posteriori error estimates; finite elements; weighted Sobolev spaces; a posteriori error estimates
UR - http://eudml.org/doc/273164
ER -

References

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