Stable mixed element schemes for plate models on multiply-connected domains

TL;DR

This paper develops stable mixed finite element schemes for Reissner-Mindlin and Kirchhoff plate models on multiply-connected domains, ensuring uniform stability through advanced decompositions and providing a general framework and specific example.

Contribution

It introduces a novel framework for designing uniformly stable mixed finite element schemes for plate models on complex domains, utilizing regular and Helmholtz decompositions.

Findings

Established equivalence of mixed and primal formulations.

Proposed a general framework for stable finite element scheme design.

Provided a specific example demonstrating the framework's application.

Abstract

In this paper, we study the mixed element schemes of the Reissner-Mindlin plate model and the Kirchhoff plate model in multiply-connected domains. By a regular decomposition of $H_0({\rm rot},\Omega)$ and a Helmholtz decomposition of its dual, we develop mixed formulations of the models which are equivalent to the primal ones respectively and which are uniformly stable. A framework of designing uniformly stable finite element schemes is presented, and a specific example is given.