Generalized mixed and primal hybrid methods with applications to plate bending

TL;DR

The paper develops a unified, DPG-inspired framework for generalized primal and mixed hybrid methods with flexible inter-element continuity, targeting self-adjoint, positive-definite operators and enabling conforming discretizations of inherently nonconforming schemes. It introduces three abstract formulations—generalized primal hybrid, generalized mixed hybrid, and ultraweak—and proves their well-posedness and quasi-optimality, supported by Fortin operator constructions and discrete inf-sup analyses. The Kirchhoff–Love plate bending model serves as the primary application, with five discretizations spanning Morley- and Zienkiewicz-type primal methods and HHJ-like mixed schemes, all attaining stable, low-order approximations and accessible trace information for bending moments and shear forces. Numerical experiments demonstrate expected convergence rates and highlight the ability to recover classical conforming behavior from nonconforming element choices, underscoring the framework’s practical relevance for plate mechanics and engineering applications.

Abstract

We present an extended framework for hybrid finite element approximations of self-adjoint, positive definite operators. It covers the cases of primal, mixed, and ultraweak formulations, both at the continuous and discrete levels, and gives rise to conforming discretizations. Our framework allows for flexible continuity restrictions across elements, and includes the extreme cases of conforming and discontinuous hybrid methods. We illustrate an application of the framework to the Kirchhoff-Love plate pending model and present three primal hybrid and two mixed hybrid methods, four of them with numerical examples. In particular, we present conforming frameworks for (in classical meaning) non-conforming elements of Morley, Zienkiewicz triangular, and Hellan-Herrmann-Johnson types.

Figures (6)

• Figure 1: Errors for the nodal-continuous primal hybrid method \ref{['KL_p2_h']}. The curves are "u": $\|u-u_h\|$, "D$^2$u": $\|D^2(u-u_h)\|_\mathcal{T}$, "Mnn": $L_2(\mathcal{S})$-error for normal-normal traces of ${\boldsymbol{M}}$, weighted with $h^{1/2}$, "shear": $L_2(\mathcal{S})$-error for effective shear force approximation, weighted with $h^{3/2}$, and curves indicating $O(h)$, $O(h^2)$.
• Figure 2: The approximation of trace component ${\boldsymbol{n}}\cdot{\boldsymbol{M}}{\boldsymbol{n}}|_\mathcal{S}$ from the nodal-continuous primal hybrid method \ref{['KL_p2_h']} (on the left) and the difference of p/w constant $L^2$-projection of ${\boldsymbol{n}}\cdot{\boldsymbol{M}}{\boldsymbol{n}}|_\mathcal{S}$ and its approximation (on the right). The mesh has 8192 elements and 12416 edges.
• Figure 3: The approximation of the effective shear force from the nodal-continuous primal hybrid method \ref{['KL_p2_h']} (absolute values, on the left) and the difference of the p/w constant $L_2$-projection of the effective shear force and its approximation (absolute values of the difference, on the right). The mesh has 8192 elements and 12416 edges.
• Figure 4: Errors for the continuous primal hybrid method \ref{['KL_p3_h']}. The curves are "u": $\|u-u_h\|_{2,\mathcal{T}}$, "Mnn": $L_2(\mathcal{S})$-error for normal-normal traces of ${\boldsymbol{M}}$, weighted with $h^{1/2}$, and a curve indicating $O(h)$.
• Figure 5: Errors for the mixed hybrid method \ref{['KL_d1_h']}. The curves are "u": $\|u-u_h\|$, "M": $\|{\boldsymbol{M}}-{\boldsymbol{M}}_h\|$, "divDiv M": $\|f-\operatorname{div}\operatorname{\mathbf{div}}{\boldsymbol{M}}_h\|_\mathcal{T}$, "D$^2$u": the $L_2$-error of the approximation of the Hessian induced by ${\boldsymbol{\psi}}_h$, along with curves indicating orders $O(h)$ and $O(h^2)$.

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 4
• Theorem 5
• Theorem 6
• Theorem 7
• proof
• Remark 8
• Theorem 9