arXiv subjects

Martin Werner Licht

Publications and source records attributed to Martin Werner Licht.

6 recordsLinked to original sources

arXiv2026

On the geometry of star domains and the spectra of Hodge-Laplace operators

We study Poincaré--Friedrichs--Weber constants for Sobolev differential forms on bounded convex domains and on domains star-shaped with respect to a ball. Generalizing work by Guerini and Savo, our main result shows that the Poincaré--Friedrichs--Weber constants in the Sobolev de~Rham complexes on bounded convex domains are nonincreasing in the degree of the differential forms. In particular, the Poincaré constant is an upper bound for the Poincaré--Friedrichs--Weber constants. We also obtain estimates for the Poincaré--Friedrichs--Weber constants on domains star-shaped with respect to a ball. As preparatory work, which may be of independent interest, we study the gauge function and the expansion function of bounded convex sets and star domains, providing new proofs of Lipschitz estimates by Vrećica and Toranzos for the expansion function and improving a Lipschitz estimate for the gauge function due to Beer.

math.AP↗

arXiv2025

Computable Poincaré--Friedrichs constants for the $L^{p}$ de~Rham complex over convex domains and domains with shellable triangulations

We construct potentials for the exterior derivative, in particular, for the gradient, the curl, and the divergence operators, over domains with shellable triangulations. Notably, the class of shellable triangulations includes local patches (stars) in two or three dimensions. The operator norms of our potentials satisfy explicitly computable bounds that depend only on the geometry. We thus compute upper bounds for constants in Poincaré--Friedrichs inequalities and lower bounds for the eigenvalues of vector Laplacians. As an additional result with independent standing, we establish Poincaré--Friedrichs inequalities with computable constants for the $L^{p}$ de~Rham complex over bounded convex domains, derived as explicit operator norms of regularized Poincaré and Bogovski\uı potential operators. We express all our main results in the calculus of differential forms and treat the gradient, curl, and divergence operators as instances of the exterior derivative. Computational examples illustrate the theoretical findings.

math.NA↗

arXiv2023

Higher-order finite element de Rham complexes, partially localized flux reconstructions, and applications

We construct finite element de~Rham complexes of higher and possibly non-uniform polynomial order in finite element exterior calculus (FEEC). Starting from the finite element differential complex of lowest-order, known as the complex of Whitney forms, we incrementally construct the higher-order complexes by adjoining exact local complexes associated to simplices. We define a commuting canonical interpolant. On the one hand, this research provides a base for studying $hp$-adaptive methods in finite element exterior calculus. On the other hand, our construction of higher-order spaces enables a new tool in numerical analysis which we call "partially localized flux reconstruction". One major application of this concept is in the area of equilibrated a~posteriori error estimators: we generalize the Braess-Schöberl error estimator to edge elements of higher and possibly non-uniform order.

math.NA↗

arXiv2023

Constructing collars in paracompact Hausdorff spaces and Lipschitz estimates

We give a constructive proof for the following new collar theorem: every locally collared closed set that is paracompact in a Hausdorff space is collared. This includes the important special case of locally collared closed sets in paracompact Hausdorff spaces. Importantly, we use Stone's result that every open cover of a paracompact space has an open locally finite refinement which is the countable union of discrete families. Furthermore, in the LIP category, our construction yields collars that are locally bi-Lipschitz embeddings. If the initial data satisfy uniform estimates, then this collar is even bi-Lipschitz onto its image and we explicitly bound the constants. We also provide partitions of unity whose Lipschitz constants are bounded by the Lebesgue constant and the order of the cover.

math.GN↗

arXiv2016

Smoothed Projections over Weakly Lipschitz Domains

We develop finite element exterior calculus over weakly Lipschitz domains. Specifically, we construct commuting projections from $L^p$ de~Rham complexes over weakly Lipschitz domains onto finite element de~Rham complexes. These projections satisfy uniform bounds for finite element spaces with bounded polynomial degree over shape-regular families of triangulations. Thus we extend the theory of finite element differential forms to polyhedral domains that are weakly Lipschitz but not strongly Lipschitz. As new mathematical tools, we use the collar theorem in the Lipschitz category, and we show that the degrees of freedom in finite element exterior calculus are flat chains in the sense of geometric measure theory.

math.NA↗

arXiv2015

Complexes of Discrete Distributional Differential Forms and their Homology Theory

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Schöberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincaré-Friedrichs-type inequalities will be studied in a subsequent contribution.

math.NA↗