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Numerische Mathematik

, Volume 137, Issue 3, pp 741–772 | Cite as

Conforming approximation of convex functions with the finite element method

  • Gerd WachsmuthEmail author
Article

Abstract

We consider the interior approximation of convex functions with convex finite element functions. The main motivation for this study is the investigation of a novel discretization of optimization problems with convexity constraints by the finite element method. Under a mild assumption on the family of meshes, we show that the conforming approximation is convergent if the finite elements are at least piecewise quadratic. We further provide similar results under additional constraints on the function values or on the gradient. The theoretical findings are illustrated by numerical examples.

Mathematics Subject Classification

65K10 65N30 

Notes

Acknowledgements

The author is indebted to the anonymous referees for many valuable comments and suggestions and for drawing his attention to [3].

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations)Technische Universität ChemnitzChemnitzGermany

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