Isoparametric finite element approximation of Ricci curvature (Q2856621)

The author uses the surface finite element method to approximate curvatures on embedded hypersurfaces and to discretize geometric partial differential equations. A definition of discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension based on the discretization of a weak formulation with isoparametric finite elements is presented. For a piecewise quadratic approximation of a two- or three-dimensional hypersurface \(\varGamma \subset \mathbb R^{n+1}\), it is proved that the stated definition approximates the Ricci curvature of \(\varGamma \) with a linear order of convergence in the \(L^{2} (\varGamma )\) norm. By using a smoothing scheme in the case of a piecewise linear approximation of \(\varGamma \), a convergence of order \(\frac{2}{3}\) in the \(L^{2} (\varGamma )\) norm and of order \(\frac{1}{3}\) in the \(W^{1,2} (\varGamma )\) norm is justified.