Finite Element Approximation of the Shape Operator via Discrete Differential Geometry
Date:
Upcoming: The slides will be provided soon.
Abstract:
Many applications, such as shell models and geometric flows, rely heavily on surface curvature. Discrete differential geometry (DDG) estimates curvature-related properties on discretized surfaces. Integrating DDG into a finite element framework provides rigorous tools for analyzing existing methods and systematically extending them. Distributional finite elements are crucial in this setup, allowing for weak regularity and requiring differential operators to be interpreted in the sense of distributions.
In this talk, we explore approximating the shape operator on surfaces described by Lagrange finite elements. Our method merges the DDG approach of dihedral angles with Hellan-Herrmann-Johnson finite elements. Using a novel integral error representation, we derive rigorous error bounds. Under mild assumptions, we show convergence in a negative Sobolev norm, thereby establishing the first proof that the dihedral-angle DDG approximation always converges. With stronger assumptions, we achieve optimal L2-convergence for the discrete L2-Riesz representative of the shape operator. We also discuss potential applications and demonstrate numerical results with the finite element library NGSolve (www.ngsolve.org). This work is in collaboration with Jay Gopalakrishnan (Portland State University).