Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary

Abstract

We establish several useful estimates for a non-conforming 2-norm posed on piecewise linear surface triangulations with boundary, with the main result being a Poincaré inequality. We also obtain equivalence of the non-conforming 2-norm posed on the true surface with the norm posed on a piecewise linear approximation. Moreover, we allow for free boundary conditions. The true surface is assumed to be C 2 , 1 C^{2,1} when free conditions are present; otherwise, C 2 C^{2} is sufficient. The framework uses tools from differential geometry and the closest point map (see [G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155]) for approximating the full surface Hessian operator. We also present a novel way of applying the closest point map when dealing with surfaces with boundary. Connections with surface finite element methods for fourth-order problems are also noted.

Finite element approximation of scalar curvature in arbitrary dimension

“…For example, for piecewise constant Regge metrics in dimension N = 2, we show that convergence of (Rω) dist (g h ) to (Rω)(g) holds in the H −2 (Ω)-norm for any optimalorder interpolant of g, but numerical experiments suggest that it fails to hold in stronger norms when g h is not the canonical interpolant of g. A key tool that we use to prove convergence in H −2 (Ω) is the near-equivalence of a certain pair of metric-dependent, mesh-dependent norms on V ; see Proposition 4.5. This equivalence is similar to one that Walker [27,Theorems 4.1 and 4.3] proved for an analogous family of mesh-dependent norms on triangulated surfaces.…”