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We present a finite element technique for approximating the surface Hessian of a discrete scalar function on triangulated surfaces embedded in $$\R^{3}$$, with or without boundary. We then extend the method to compute approximations of the full shape operator of the underlying surface using only the known discrete surface. The method is based on the Hellan--Herrmann--Johnson (HHJ) element and does not require any ad-hoc modifications. Convergence is established provided the discrete surface satisfies a Lagrange interpolation property related to the exact surface. The convergence rate, in $L^2$, for the shape operator approximation is $O(h^m)$, where $$m \geq 1$$ is the polynomial degree of the surface, i.e. the method converges even for piecewise linear surface triangulations. For surfaces with boundary, some additional boundary data is needed to establish optimal convergence, e.g. boundary information about the surface normal vector or the curvature in the co-normal direction. Numerical examples are given on non-trivial surfaces that demonstrate our error estimates and the efficacy of the method.
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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.
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- Award ID(s):
- 2111474
- PAR ID:
- 10343184
- Date Published:
- Journal Name:
- Computational Methods in Applied Mathematics
- Volume:
- 22
- Issue:
- 1
- ISSN:
- 1609-4840
- Page Range / eLocation ID:
- 227 to 243
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/cmam-2020-0123
