Conforming approximation of convex functions with the finite element method

Wachsmuth, Gerd

doi:10.1007/s00211-017-0884-8

Cited by 8 publications

(3 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is important to realize that this definition does not imply convexity in the usual sense, which is rather tricky to achieve with piecewise polynomials [1,32,37]. On the other hand, if w ∈ C 0 (Ω h ) is convex, then its Lagrange interpolant I h w satisfies I h w ≥ w, whence I h w is discretely convex but not necessarily convex.…”

Section: Two-scale Methodsmentioning

confidence: 99%

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

2018

View full text Add to dashboard Cite

We propose a two-scale finite element method for the Monge-Ampère equation with Dirichlet boundary condition in dimension d ≥ 2 and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite difference method of Froese and Oberman, but is defined on unstructured grids and relies on two separate scales: the first one is the mesh size h and the second one is a larger scale that controls appropriate directions and substitutes the need of a wide stencil. The main tools for the analysis are a discrete comparison principle and discrete barrier functions that control the behavior of the discrete solution, which is continuous piecewise linear, both close to the boundary and in the interior of the domain.

show abstract

Section: Two-scale Methodsmentioning

confidence: 99%

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

2018

View full text Add to dashboard Cite

show abstract

“…Indeed, in two dimensions it is not possible to approximate arbitrary convex functions by convex finite element functions of order one on, e.g., uniform refinements of a fixed grid, see [Choné, Le Meur, 2001]. However, for functions of order at least two, there exist approximation results, see [Aguilera, Morin, 2009;Wachsmuth, 2017]. the rotational symmetry of the problem is strongly violated.…”

Section: Non-convergence In 3 Dimensionsmentioning

confidence: 99%

Numerical Approximation of Optimal Convex Shapes

Bartels¹,

Wachsmuth²

2020

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem. Moreover, we prove the convergence of discretizations in two-dimensional situations. A numerical algorithm is devised that iteratively solves the discrete formulation. Numerical experiments show that optimal convex shapes are generally non-smooth and that three-dimensional problems require an appropriate relaxation of the convexity condition.

show abstract

“…To approximate convex functions, we need for example higher order conforming finite elements (c.f. [32]), a weaker definition for convexity tailored to finite elements (c.f. [1]), a geometric approach as in [26] or spherical harmonic decomposition (c.f.…”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation

Keller,

Bartels,

Wachsmuth

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in R 3 , the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.

show abstract

Conforming approximation of convex functions with the finite element method

Cited by 8 publications

References 29 publications

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

Numerical Approximation of Optimal Convex Shapes

Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation

Contact Info

Product

Resources

About