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.

Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

“…In a later section, we extend the ideas in [27,11] by exploiting certain orthogonality properties for the error in the canonical interpolation by Regge finite elements to obtain one extra order of convergence for the curvature approximation. This extra order is comparable to super-convergence properties of mixed methods [12,22] and has been observed for the Hellan-Herrmann-Johnson for the biharmonic plate and shell equation [9,46].…”

“…in the lowest order case, k = 0, linear convergence, although a second order differential operator relying on derivatives of the metric tensor is applied. The idea of splitting the terms into a high-order and a sub-optimal one, which then turns out to be zero, has been used in [46]. Therein, the convergence of the surface divdiv operator on an approximated triangulation is proven in an extrinsic manner to converge optimally.…”

Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

“…In a later section, we extend the ideas in [10,27] by exploiting certain orthogonality properties for the error in the canonical interpolation by Regge finite elements to obtain one extra order of convergence for the curvature approximation. This extra order is comparable to super-convergence properties of mixed methods [11,21] and has been observed for the Hellan-Herrmann-Johnson method for the biharmonic plate and shell equation [8,48]. The heart of the matter are FEEC-type identities that show that the error in canonical interpolations superconverges.…”

“…Also notable from our analysis so far is the idea of splitting the error terms into a high-order ones and ones that might be sub-optimal in general, but vanishes in specific cases. Such an idea was also used in [48], where the convergence of a surface div div operator on an approximated triangulation is proven (in an extrinsic manner) to converge.…”