Search for: All records

The Whitney forms on a simplex T admit high-order generalizations that have received a great deal of attention in numerical analysis. Less well-known are the shadow forms of Brasselet, Goresky, and MacPherson. These forms generalize the Whitney forms, but have rational coefficients, allowing singularities near the faces of T. Motivated by numerical problems that exhibit these kinds of singularities, we introduce degrees of freedom for the shadow k-forms that are well-suited for finite element implementations. In particular, we show that the degrees of freedom for the shadow forms are given by integration over the k-dimensional faces of the blow-up T̃ of the simplex T. Consequently, we obtain an isomorphism between the cohomology of the complex of shadow forms and the cellular cohomology of T̃, which vanishes except in degree zero. Additionally, we discover a surprising probabilistic interpretation of shadow forms in terms of Poisson processes. This perspective simplifies several proofs and gives a way of computing bases for the shadow forms using a straightforward combinatorial calculation. 

We analyze finite element discretizations of scalar curvature in dimension $$N \ge 2$$. Our analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric $$g$$ on a simplicial triangulation of a polyhedral domain $$\Omega \subset \mathbb{R}^N$$ having maximum element diameter $$h$$. We show that if such an interpolant $$g_h$$ has polynomial degree $$r \ge 0$$ and possesses single-valued tangential-tangential components on codimension-1 simplices, then it admits a natural notion of (densitized) scalar curvature that converges in the $$H^{-2}(\Omega)$$-norm to the (densitized) scalar curvature of $$g$$ at a rate of $$O(h^{r+1})$$ as $$h \to 0$$, provided that either $N = 2$ or $$r \ge 1$$. As a special case, our result implies the convergence in $$H^{-2}(\Omega)$$ of the widely used ``angle defect'' approximation of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on the interpolated metric $$g_h$$. We present numerical experiments that indicate that our analytical estimates are sharp. 

We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of total energy and magnetic helicity, and preserves mass and the constraint $$\text {div}B = 0$$ to machine precision, both at the spatially and temporally discrete levels. In particular, conservation of energy and magnetic helicity hold at the discrete levels in the ideal case. It is observed that cross-helicity is well conserved in our simulation in the ideal case. 

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.