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

This is the authors' abstract. We don't add key points for this paper.

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 C2,1 C2,1 when free conditions are present; otherwise, C2 C² 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.