arXiv subjects

Martin W. Licht

Publications and source records attributed to Martin W. Licht.

9 recordsLinked to original sources

arXiv2024

On Lipschitz partitions of unity and the Assouad--Nagata dimension

We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant $\max(1,M-1)/\mathcal{L}$, where $\mathcal{L}$ is the Lebesgue number and $M$ is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, such as length spaces do, then the upper bound improves to $(M-1)/(2\mathcal{L})$. These Lipschitz estimates are optimal. We also address the Lipschitz analysis of $\ell^{p}$-generalizations of the standard partition of unity, their partial sums, and their categorical products. Lastly, we characterize metric spaces with Assouad--Nagata dimension $n$ as exactly those metric spaces for which every Lebesgue cover admits an open refinement with multiplicity $n+1$ while reducing the Lebesgue number by at most a constant factor.

math.MG↗

arXiv2023

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 within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on uniformly bounded commuting projections on manifolds: these projections map from Sobolev de Rham complexes onto finite element de Rham complexes, commute with the differential operators, and satisfy uniform bounds in Lebesgue norms. They enable the Galerkin theory of Hilbert complexes for a large range of intrinsic finite element methods on manifolds. However, these intrinsic finite element methods are generally not computable and thus primarily of theoretical interest. This leads to our second point: estimating the geometric variational crime incurred by transitioning to computable approximate problems. Lastly, our third point addresses how to estimate the approximation error of the intrinsic finite element method in terms of the mesh size. If the solution is not continuous, then such an estimate is achieved via modified Clément or Scott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.

math.NA↗

arXiv2023

Smoothed projections over manifolds in finite element exterior calculus

We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on manifolds. The commuting projections use localized mollification operators, building upon a classical construction by de Rham. These projections are uniformly bounded on Lebesgue spaces of differential forms and map onto intrinsic finite element spaces defined with respect to an intrinsic smooth triangulation of the manifold. We analyze the Galerkin approximation error. Since practical computations use extrinsic finite element methods over approximate computational manifolds, we also analyze the geometric error incurred.

math.NA↗

arXiv2023

Symmetry and Invariant Bases in Finite Element Exterior Calculus

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 indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan's classification of monomial irreducible representations of the symmetric group, and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms.

math.NA↗

arXiv2023

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

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 determine sufficient regularity conditions for the coordinate change. We also introduce a novel higher-order chain rule for composition chains of multivariate functions that is described via nested set partitions and generalized Bell polynomials; it is a natural extension of the Faà di Bruno formula. Our discussion uses the coordinate-free language of tensor calculus and includes Fréchet-differentiable mappings between Banach spaces.

math.FA↗

arXiv2023

Averaging-based local projections in finite element exterior calculus

We develop projection operators onto finite element differential forms over simplicial meshes. Our projection is locally bounded in Lebesgue and Sobolev-Slobodeckij norms, uniformly with respect to mesh parameters. Moreover, it incorporates homogeneous boundary conditions and satisfies a local broken Bramble-Hilbert estimate. The construction principle includes the Ern-Guermond projection and a modified Clément-type interpolant with the projection property. The latter seems to be a new result even for Lagrange elements. This projection operator immediately enables an equivalence result on local- and global-best approximations. We combine techniques for the Scott-Zhang and Ern-Guermond projections and adopt the framework of finite element exterior calculus. We instantiate the abstract projection for Brezzi-Douglas-Marini, Nédélec, and Raviart-Thomas elements.

math.NA↗

arXiv2022

On Basis Constructions in Finite Element Exterior Calculus

We give a systematic and self-contained account of the construction of 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 element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or give a new derivation of the degrees of freedom.

math.NA↗

arXiv2020

Local Finite Element Approximation of Sobolev Differential Forms

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.

math.NA↗

arXiv2017

Smoothed Projections and Mixed Boundary Conditions

Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve homogeneous boundary conditions along a fixed boundary part, and satisfy uniform bounds for shape-regular families of triangulations and bounded polynomial degree. The existence of such projections implies stability and quasi-optimal convergence of mixed finite element methods for the Hodge Laplace equation with mixed boundary conditions. In addition, we prove the density of smooth differential forms in Sobolev spaces of differential forms over weakly Lipschitz domains with partial boundary conditions.

math.NA↗