On Locking and Robustness in the Finite Element Method

Abstract

A numerical scheme for the approximation of a parameter-dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value. A robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter. Precise mathematical definitions for these terms are developed, their quantitative characterization is given, and some general theorems involving locking and robustness are proven. A model problem involving heat transfer is analyzed in detail using this mathematical framework, and various related computational results are described. Applications to some different problems involving locking are presented.