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Abstract We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in $${\mathbb {R}}^{3}$$, with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for $$C^{k+1}$$ surfaces, with degree $$k$$ (Lagrangian, parametrically curved) approximation of the surface, for any $$k \geqslant 1$$. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least $$C^{2,1}$$. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations, vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method.
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Approximating the Shape Operator with the Surface Hellan–Herrmann–Johnson Element
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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- Award ID(s):
- 2111474
- PAR ID:
- 10524443
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM Journal on Scientific Computing
- Volume:
- 46
- Issue:
- 2
- ISSN:
- 1064-8275
- Page Range / eLocation ID:
- A1252 to A1275
- Subject(s) / Keyword(s):
- surface Hessian, shape operator, surface finite elements, open surfaces, geometric consistency error
- 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/10.1137/22M1531968
