Isoparametric finite element approximation of Ricci curvature

Abstract

The surface finite element method can be used to approximate curvatures on embedded hypersurfaces and to discretize geometric partial differential equations. In this paper, we present a definition of discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension based on the discretization of a weak formulation with isoparametric finite elements. We prove that for a piecewise quadratic approximation of a two- or three-dimensional hypersurface Γ ⊂ ℝn+1, this definition approximates the Ricci curvature of Γ with a linear order of convergence in the L2 (Γ) norm. By using a smoothing scheme in the case of a piecewise linear approximation of Γ, we still get a convergence of order ⅔ in the L2 (Γ) norm and of order ⅓ in the W1, 2 (Γ) norm.