Publications de Martin Werner Licht

Towards Finite Element Exterior Calculus on Manifolds: Commuting Projections, Geometric Variational Crimes, and Approximation Errors

We survey recent contributions to finite element exterior calculus on manifolds and surfaces, giving a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. We first discuss uniformly bounded commuting ...

Springer Science and Business Media Deutschland GmbH2025

On Lipschitz partitions of unity and the Assouad-Nagata dimension

Martin Werner Licht

We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant max(1, M - 1)/L, where Lis the Lebesgue number and M is the multiplicity of the cover. If the metric space satisfies the approximate midpoint ...

ELSEVIER2024

Symmetry and Invariant Bases in Finite Element Exterior Calculus

Martin Werner Licht

We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex ...

2023

Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and total variation spaces

Martin Werner Licht

We describe higher-order chain rules for multivariate functions and tensor fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the total variation seminorms of the higher derivatives of tensor fields after a change of variables and de ...

2023

Geometric transformation of finite element methods: Theory and applications

Martin Werner Licht

We present a new analysis of finite element methods for partial differential equations over curved domains. In many applications, a change of variables translates a physical Poisson problem over a curved physical domain into a parametric Poisson problem ov ...

2023

On basis constructions in finite element exterior calculus

Martin Werner Licht

We give a systematic self-contained exposition of how to construct geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite e ...

2022