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Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω)

Published online by Cambridge University Press:  04 July 2014

Thomas Apel ,
Ariel L. Lombardi  and
Max Winkler
Thomas Apel
Affiliation:
Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Germany. thomas.apel@unibw.de; max.winkler@unibw.de
Ariel L. Lombardi
Affiliation:
Departamento de Matemática, Universidad de Buenos Aires, and Instituto de Ciencias, Universidad Nacional de General Sarmiento. Member of CONICET, Argentina; aldoc7@dm.uba.ar
Max Winkler
Affiliation:
Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Germany. thomas.apel@unibw.de; max.winkler@unibw.de
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Abstract

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equations and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.

Keywords

Elliptic boundary value problemedge and vertex singularitiesfinite element methodanisotropic mesh gradingoptimal control problemdiscrete compactness property
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1117 - 1145
Copyright
© EDP Sciences, SMAI 2014

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