Search for: All records

Abstract We construct and analyse finite element approximations of the Einstein tensor in dimension $$N \ge 3$$. We focus on the setting where a smooth Riemannian metric tensor $$g$$ on a polyhedral domain $$\varOmega \subset \mathbb{R}^{N}$$ has been approximated by a piecewise polynomial metric $$g_{h}$$ on a simplicial triangulation $$\mathcal{T}$$ of $$\varOmega $$ having maximum element diameter $$h$$. We assume that $$g_{h}$$ possesses single-valued tangential–tangential components on every codimension-$$1$$ simplex in $$\mathcal{T}$$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $$g_{h}$$ to the Einstein curvature of $$g$$ under refinement of the triangulation. We show that in the $$H^{-2}(\varOmega )$$-norm this convergence takes place at a rate of $$O(h^{r+1})$$ when $$g_{h}$$ is an optimal-order interpolant of $$g$$ that is piecewise polynomial of degree $$r \ge 1$$. We provide numerical evidence to support this claim. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric. 

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.