Inf-sup condition and locking: Understanding and circumventing. Stokes, Laplacian, bi-Laplacian, Kirchhoff--Love and Mindlin--Reissner locking type, boundary conditions

TL;DR

This paper explains the inf-sup condition's role in finite element methods, analyzes why it fails for certain discretizations, and reviews strategies to overcome such locking issues across various PDEs.

Contribution

It provides an engineer-friendly explanation of inf-sup failure causes and reviews existing methods to circumvent locking in finite element discretizations.

Findings

Identifies reasons for inf-sup condition failure in finite element discretizations.

Reviews current techniques to address locking issues.

Connects mathematical foundations with practical engineering approaches.

Abstract

The inf-sup condition, also called the Ladyzhenskaya--Babu\v ska--Brezzi (LBB) condition, ensures the existence, uniqueness and well-posedness of a saddle point problem, relative to a partial differential equation. Discretization by the finite element method gives the discrete problem which must satisfy the discrete inf-sup condition. But, depending on the choice of finite elements, the discrete condition may fail. This paper attempts to explain why it fails from an engineer's perspective, and reviews current methods to work around this failure. The last part recalls the mathematical bases.