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Texas A&M University
Mathematics

Numerical Analysis Seminar

Date: November 16, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 306

Speaker: Shawn Walker, Louisiana State University

  

Title: Curvature and the HHJ Method

Abstract: This talk shows how the classic Hellan--Herrmann--Johnson (HHJ) method can be extended to surfaces, as well as approximate curvature. We start by showing how HHJ can be extended to surfaces embedded in ℝ3 to solve the surface version of the Kirchhoff plate equation. The surface Hessian of the "displacement" variable is discretized by an HHJ finite element function. Convergence is established for all possible combinations of mixed boundary conditions, e.g. clamped, simply-supported, free, and the "4th" condition. Numerical examples are shown, some which use "point" boundary conditions as well as solving the surface biharmonic equation.

We also show how the surface HHJ method can be used to post-process a discrete Lagrange function on a given surface triangulation to yield an approximation of its surface Hessian, i.e. a kind of Hessian recovery. Moreover, we demonstrate that this scheme can be used to give convergent approximations of the *full shape operator* of the underlying surface using only the known discrete surface, even for piecewise linear triangulations. Several numerical examples are given on non-trivial surfaces that demonstrate the scheme.