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

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 Sch\"oberl, 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. Poincar\'e-Friedrichs-type inequalities will be studied in a subsequent contribution.Comment: revised preprint, 26 page

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 Fa\`a di Bruno formula. Our discussion uses the coordinate-free language of tensor calculus and includes Fr\'echet-differentiable mappings between Banach spaces.Comment: Submitte

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. The group of affine symmetries of a simplex is isomorphic to a permutation group and represented on simplicial finite element spaces by the pullback action. We want to know which vector-valued finite element spaces have bases that are invariant under permutation of vertex indices. 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 Djokovic and Malzan's classification of monomial irreducible representations of the symmetric group and use symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Invariant bases are constructed in dimensions two and three for different spaces of finite element differential forms.Comment: 27 pages. Submitte

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.Comment: Submitted. 31 page

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.Comment: Feedback welcom

LOCOMOTOR FUNCTION OF SCALES AND AXIAL SKELETON IN MIDDLE–LATE TRIASSIC SPECIES OF SAURICHTHYS (ACTINOPTERYGII)

Starting in the Late Permian, the “Triassic osteichthyan revolution” gave rise to several new morphotypes of actinopterygians, including the iconic barracuda-shaped predator Saurichthys. About 50 species, from 10 cm to over 1.5 m long, are known from mainly marine deposits worldwide. Despite current interest in Saurichthys, freshwater species and those from late Middle to early Late Triassic remain understudied. We document the postcranial morphology of three small to mid-sized (15–45 cm) species from this timeframe represented by sufficiently complete individuals: Saurichthys orientalis Sytchevskaya, 1999, from lacustrine deposits of the Madygen Formation (late Ladinian/Carnian); S. striolatus (Bronn, 1858) from the fully marine Predil Limestone (early Carnian); and S. calcaratus Griffith, 1977, from the terrigenously influenced coastal environment of the Lunz Formation (middle Carnian). S. orientalis resembles early saurichthyids in having six rows of large, thick ganoid scales; fins with segmented lepidotrichia; and flank scales relating to dorsal vertebral elements as 1:2. S. calcaratus and S. striolatus share unsegmented fin rays and a reduced scale cover with well-ossified but narrow mid-dorsal and mid-ventral scales and small, thin flank scales, relating to the dorsal arcualia as 1:1. Ventral arcualia are first described for S. calcaratus and S. striolatus, where they change in shape and number at the abdominal-caudal transition. In all three species, force transmission to the tail fin is enhanced by the caudal peduncle strengthened by a stiff structure arising from interlocking or fusion of the last enlarged mid-dorsal and mid-ventral scales (scutes), while the vertebral column remains rather lightly built