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IMA Journal of Numerical Analysis Advance Access originally published online on July 30, 2008
IMA Journal of Numerical Analysis 2009 29(4):917-936; doi:10.1093/imanum/drn039
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Convergence rates for adaptive finite elements

Fernando D. Gaspoz{dagger} and Pedro Morin{ddagger}

Instituto de Matemática Aplicada del Litoral, Consejo Nacional de Investigaciones Científicas y Técnicas and Universidad Nacional del Litoral, Güemes 3450, S3000GLN Santa Fe, Argentina

{dagger} Corresponding author. Email: fgaspoz{at}santafe-conicet.gov.ar

{ddagger} Email: pmorin{at}santafe-conicet.gov.ar

Received on 6 November 2007. Revised on 18 April 2008.


  Abstract

In this article, we prove that it is possible to construct, using newest vertex bisection, meshes that equidistribute the error in the H1-norm whenever the function to be approximated can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of partial differential equations. As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite-element methods using Lagrange finite elements of any polynomial degree are obtained.

Key Words: convergence rates; adaptive finite elements; optimality; regularity


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