Generalized mixed and primal hybrid methods with applications to plate bending

TL;DR

This paper develops a comprehensive hybrid finite element framework applicable to various formulations of self-adjoint operators, enabling flexible discretizations and including conforming and non-conforming methods, demonstrated through plate bending applications.

Contribution

It introduces a unified framework for hybrid finite element methods covering primal, mixed, and ultraweak formulations with flexible continuity, including new conforming approaches for non-conforming elements.

Findings

Developed a versatile hybrid finite element framework.

Presented conforming discretizations for non-conforming elements.

Demonstrated applications to Kirchhoff-Love plate models.

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.