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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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The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method
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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- Award ID(s):
- 2111474
- PAR ID:
- 10343187
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- ISSN:
- 0272-4979
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1093/imanum/drab062
