# Convex hull property and maximum principle for finite element minimisers of general convex functionals

@article{Diening2013ConvexHP, title={Convex hull property and maximum principle for finite element minimisers of general convex functionals}, author={Lars Diening and Christian Kreuzer and Sebastian Schwarzacher}, journal={Numerische Mathematik}, year={2013}, volume={124}, pages={685-700} }

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 $$\mathbb{P }_1$$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly…

## 24 Citations

### On the positivity of discrete harmonic functions and the discrete Harnack inequality for piecewise linear finite elements

- MathematicsMath. Comput.
- 2017

It is established that in two dimensions on a smooth domain the discrete Green’s function with singularity in the interior of the domain must be strictly positive throughout the computational domain once the mesh is sufficiently refined.

### Discrete comparison principles for quasilinear elliptic PDE

- MathematicsApplied Numerical Mathematics
- 2020

### Partial Regularity for BV Minimizers

- MathematicsArchive for Rational Mechanics and Analysis
- 2018

We establish an $${\varepsilon}$$ε-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a…

### The Regularity of Minima for the Dirichlet Problem on BD

- MathematicsArchive for Rational Mechanics and Analysis
- 2020

We establish that the Dirichlet problem for linear growth functionals on $${\text {BD}}$$ BD , the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial $${\text…

### A note on why enforcing discrete maximum principles by a simple a posteriori cutoff is a good idea

- Mathematics, Computer Science
- 2014

This work enforce the discrete maximum principle by performing a simple cutoff and shows that for many problems this a posteriori procedure even improves the approximation in the natural energy norm.

### Uniform Hölder-norm bounds for finite element approximations of second-order elliptic equations

- MathematicsIMA Journal of Numerical Analysis
- 2020

We develop a discrete counterpart of the De Giorgi–Nash–Moser theory, which provides uniform Hölder-norm bounds on continuous piecewise affine finite element approximations of second-order linear…

### Some discrete maximum principles arising for nonlinear elliptic finite element problems

- MathematicsComput. Math. Appl.
- 2015

### N A ] 3 1 O ct 2 01 7 DISCRETE COMPARISON PRINCIPLES FOR QUASILINEAR ELLIPTIC PDE

- Mathematics
- 2021

Comparison principles are developed for discrete quasilinear elliptic partial differential equations. We consider the analysis of a class of nonmonotone LerayLions problems featuring both nonlinear…

### A property for the Monge-Ampère equation

- Mathematics
- 2020

Let Ω ⊆ ℝ n be a non-empty open bounded set and h : Ω → ℝ be a non-negative continuous function. We prove that for any u ∈ C 2 (Ω) ∩ C 1 ( $$\overline{\Omega}$$ ) solution of the Monge–Ampère…

### Partial regularity of minimizers of functionals with discontinuous coefficients of low integrability with applications to nonlinear elliptic systems

- MathematicsCommunications in Partial Differential Equations
- 2018

Abstract In this article, we consider minimizers of the functional where is open and bounded and . We show that even though the coefficient a satisfies only that , it nonetheless follows that u is…

## References

SHOWING 1-10 OF 29 REFERENCES

### Maximum Principles for P1-Conforming Finite Element Approximations of Quasi-linear Second Order Elliptic Equations

- MathematicsSIAM J. Numer. Anal.
- 2012

This paper derives some discrete maximum principles for $P1-conforming finite element approximations for quasi-linear second order elliptic equations from the classical maximum principles in the theory of partial differential equations to finite element methods.

### Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations

- MathematicsNumerische Mathematik
- 2005

Summary.Uniform lower and upper bounds for positive finite-element approximations to semilinear elliptic equations in several space dimensions subject to mixed Dirichlet-Neumann boundary conditions…

### Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

- MathematicsNumerische Mathematik
- 2005

This work presents several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions and some examples of real-life problems, where the preservation of maximum principles plays an important role.

### The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem

- Mathematics
- 2008

### Failure of the discrete maximum principle for an elliptic finite element problem

- MathematicsMath. Comput.
- 2005

This work studies the question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem via the Galerkin method with continuous piecewise linears, and extends the number of cases where it is known to be positive.

### Global and local refinement techniques yielding nonobtuse tetrahedral partitions

- Mathematics, Computer Science
- 2005

### Partial regularity for a class of anisotropic variational integrals with convex hull property

- Mathematics
- 2001

We consider integrands f:\mathbb{R}^{nN}\rightarrow\mathbb{R} which are of lower (upper) growth rate s\geq2(q>s) and which satisfy an additional structural condition implying the convex hull…

### Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle

- MathematicsMath. Comput.
- 2001

We prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the Poisson equation with the Dirichlet boundary condition also under a condition of…

### Three-Dimensional Delaunay Triangulations for Finite Element Approximations to a Second-Order Diffusion Operator

- MathematicsSIAM J. Sci. Comput.
- 1992

It is shown that a three-dimensional Delaunay triangulation does not generally produce a discretisation satisfying the condition of positive interior connection condition, and it is generally not possible to produce aThree-dimensional triangulations that satisfies the positive interior connected condition.