Conforming approximation of convex functions with the finite element method

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 o...

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Bibliographic Details
Published inNumerische Mathematik Vol. 137; no. 3; pp. 741 - 772
Main Author Wachsmuth, Gerd
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2017
Springer Nature B.V
Subjects
Approximation
Convex analysis
Convexity
Finite element analysis
Finite element method
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Nonlinear programming
Numerical Analysis
Numerical and Computational Physics
Simulation
Theoretical
65K10
65N30
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Summary: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.
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-017-0884-8