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

Thomas Apel; Ariel L. Lombardi; Max Winkler

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

  • Volume: 48, Issue: 4, page 1117-1145
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Apel, Thomas, Lombardi, Ariel L., and Winkler, Max. "Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω)." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.4 (2014): 1117-1145. <http://eudml.org/doc/273100>.

@article{Apel2014,
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.},
author = {Apel, Thomas, Lombardi, Ariel L., Winkler, Max},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {elliptic boundary value problem; edge and vertex singularities; finite element method; anisotropic mesh grading; optimal control problem; discrete compactness property; error estimates; Poisson equation; quasi-interpolantion operator; numerical result},
language = {eng},
number = {4},
pages = {1117-1145},
publisher = {EDP-Sciences},
title = {Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω)},
url = {http://eudml.org/doc/273100},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Apel, Thomas
AU - Lombardi, Ariel L.
AU - Winkler, Max
TI - Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω)
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 4
SP - 1117
EP - 1145
AB - 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.
LA - eng
KW - elliptic boundary value problem; edge and vertex singularities; finite element method; anisotropic mesh grading; optimal control problem; discrete compactness property; error estimates; Poisson equation; quasi-interpolantion operator; numerical result
UR - http://eudml.org/doc/273100
ER -

References

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