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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 beC2,1C^{2,1}when free conditions are present; otherwise,C2C^{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.

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