# Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals

Diening L, Kreuzer C, Schwarzacher S (2012)
SIAM Journal on Mathematical Analysis 44(5): 3594-3616.

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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 often crucial for the preservation of qualitative properties of the respective 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 minimiser 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 introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.
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Diening L, Kreuzer C, Schwarzacher S. Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals. SIAM Journal on Mathematical Analysis. 2012;44(5):3594-3616.
Diening, L., Kreuzer, C., & Schwarzacher, S. (2012). Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals. SIAM Journal on Mathematical Analysis, 44(5), 3594-3616. doi:10.1137/120870554
Diening, L., Kreuzer, C., and Schwarzacher, S. (2012). Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals. SIAM Journal on Mathematical Analysis 44, 3594-3616.
Diening, L., Kreuzer, C., & Schwarzacher, S., 2012. Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals. SIAM Journal on Mathematical Analysis, 44(5), p 3594-3616.
L. Diening, C. Kreuzer, and S. Schwarzacher, “Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals”, SIAM Journal on Mathematical Analysis, vol. 44, 2012, pp. 3594-3616.
Diening, L., Kreuzer, C., Schwarzacher, S.: Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals. SIAM Journal on Mathematical Analysis. 44, 3594-3616 (2012).
Diening, Lars, Kreuzer, Christian, and Schwarzacher, Sebastian. “Convex Hull Property and Maximum Principle for Finite Element Minimizers of General Convex Functionals”. SIAM Journal on Mathematical Analysis 44.5 (2012): 3594-3616.
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arXiv 1302.0112