Sparsity optimized high order finite element functions for H(curl) on tetrahedra

AbstractH(curl) conforming finite element discretizations are a powerful tool for the numerical solution of the system of Maxwellʼs equations in electrodynamics. In this paper we construct a basis for conforming high-order finite element discretizations of the function space H(curl) in 3 dimensions. We introduce a set of hierarchic basis functions on tetrahedra with the property that both the L2-inner product and the H(curl)-inner product are sparse with respect to the polynomial degree. The construction relies on a tensor-product based structure with properly weighted Jacobi polynomials as well as an explicit splitting of the basis functions into gradient and non-gradient functions. The basis functions yield a sparse system matrix with O(1) nonzero entries per row.The proof of the sparsity result on general tetrahedra defined in terms of their barycentric coordinates is carried out by an algorithm that we implemented in Mathematica. A rewriting procedure is used to explicitly evaluate the inner products. The precomputed matrix entries in this general form for the cell-based basis functions are available online

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

Semantic 3D Reconstruction with Finite Element Bases

We propose a novel framework for the discretisation of multi-label problems on arbitrary, continuous domains. Our work bridges the gap between general FEM discretisations, and labeling problems that arise in a variety of computer vision tasks, including for instance those derived from the generalised Potts model. Starting from the popular formulation of labeling as a convex relaxation by functional lifting, we show that FEM discretisation is valid for the most general case, where the regulariser is anisotropic and non-metric. While our findings are generic and applicable to different vision problems, we demonstrate their practical implementation in the context of semantic 3D reconstruction, where such regularisers have proved particularly beneficial. The proposed FEM approach leads to a smaller memory footprint as well as faster computation, and it constitutes a very simple way to enable variable, adaptive resolution within the same model

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