The Mathematical Institute, University of Oxford, Eprints Archive

Convex hull property and maximum principles for finite element minimizers of general convex functionals

Diening, Lars and Kreuzer, Christian and Schwarzacher, Sebastian (2012) Convex hull property and maximum principles for finite element minimizers of general convex functionals. Technical Report. Numerische Mathematik. (Submitted)

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Abstract

The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are crucial for the preservation of qualitative properties of the physical model. In this work we develop a convex hull property for $P_{1}$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimizer of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the $p$-Laplacian and the mean curvature problem. In the case of scalar equations the presented arguments can be used to prove standard discrete maximum principles for nonlinear problems.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1501
Deposited By:Lotti Ekert
Deposited On:27 Mar 2012 07:49
Last Modified:27 Mar 2012 07:49

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