Interpolation of non-smooth functions on anisotropic finite element meshes

Thomas Apel

doi:10.1051/m2an:1999139

Abstract

In this paper, several modifications of the quasi-interpolationoperator of Scott andZhang [30]are discussed. The modified operators are defined for non-smooth functionsand are suited for application on anisotropic meshes. Theanisotropy of the elements is reflected in the local stability andapproximation error estimates.As an application, an example is considered where anisotropic finiteelement meshes are appropriate, namely the Poisson problem in domainswith edges.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Carstensen, Carsten and Funken, Stefan A. 1999. Fully Reliable Localized Error Control in the FEM. SIAM Journal on Scientific Computing, Vol. 21, Issue. 4, p. 1465.

Kunert, Gerd 2001. A Local Problem Error Estimator for Anisotropic Tetrahedral Finite Element Meshes. SIAM Journal on Numerical Analysis, Vol. 39, Issue. 2, p. 668.

Kunert, Gerd 2002. Toward anisotropic mesh construction and error estimation in the finite element method. Numerical Methods for Partial Differential Equations, Vol. 18, Issue. 5, p. 625.

Yueqiang Liu Bondeson, A. Bergstrom, R. Larson, M.G. and Samuelsson, K. 2003. Edge element computations of eddy currents in laminated materials. IEEE Transactions on Magnetics, Vol. 39, Issue. 3, p. 1758.

Apel, Thomas 2004. Encyclopedia of Computational Mechanics.

Picasso, M. 2005. An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio. Applied Numerical Mathematics, Vol. 54, Issue. 3-4, p. 470.

Picasso, Marco 2006. Anisotropica posteriori error estimate for an optimal control problem governed by the heat equation. Numerical Methods for Partial Differential Equations, Vol. 22, Issue. 6, p. 1314.

Nicaise, Serge and Sirch, Dieter 2011. Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity. Computational Optimization and Applications, Vol. 49, Issue. 3, p. 567.

Apel, Thomas and Sirch, Dieter 2011. L 2-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes. Applications of Mathematics, Vol. 56, Issue. 2, p. 177.

Apel, Thomas Flaig, Thomas G. and Nicaise, Serge 2014. A Priori Error Estimates for Finite Element Methods forH(2,1)-Elliptic Equations. Numerical Functional Analysis and Optimization, Vol. 35, Issue. 2, p. 153.

Apel, Thomas Lombardi, Ariel L. and Winkler, Max 2014. Anisotropic mesh refinement in polyhedral domains: error estimates with data inL2(Ω). ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 48, Issue. 4, p. 1117.

Karkulik, M. and Melenk, J.M. 2015. Local high-order regularization and applications to hp-methods. Computers & Mathematics with Applications, Vol. 70, Issue. 7, p. 1606.

Langer, Ulrich Mantzaflaris, Angelos Moore, Stephen E. and Toulopoulos, Ioannis 2015. Mesh grading in isogeometric analysis. Computers & Mathematics with Applications, Vol. 70, Issue. 7, p. 1685.

Nochetto, Ricardo H. Otárola, Enrique and Salgado, Abner J. 2015. A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis. Foundations of Computational Mathematics, Vol. 15, Issue. 3, p. 733.

Nochetto, Ricardo H. Otárola, Enrique and Salgado, Abner J. 2016. Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numerische Mathematik, Vol. 132, Issue. 1, p. 85.

Apel, Thomas and Melenk, Jens M. 2017. Encyclopedia of Computational Mechanics Second Edition. p. 1.

Banjai, Lehel Melenk, Jens M. Nochetto, Ricardo H. Otárola, Enrique Salgado, Abner J. and Schwab, Christoph 2019. Tensor FEM for Spectral Fractional Diffusion. Foundations of Computational Mathematics, Vol. 19, Issue. 4, p. 901.

Toulopoulos, Ioannis 2020. Space-time finite element methods stabilized using bubble function spaces. Applicable Analysis, Vol. 99, Issue. 7, p. 1153.

Faustmann, Markus Melenk, Jens Markus and Parvizi, Maryam 2021. On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 55, Issue. 2, p. 595.

Mauri, Aurelio Giancarlo Morini, Benedetta Perotto, Simona and Sgallari, Fiorella 2023. Springer Handbook of Semiconductor Devices. p. 1383.

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Interpolation of non-smooth functions on anisotropic finite element meshes

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