Locking and robustness in the finite element method for circular arch problem

Summary.

In this paper we discuss locking and robustness of the finite element method for a model circular arch problem. It is shown that in the primal variable (i.e., the standard displacement formulation), the p-version is free from locking and uniformly robust with order$p^{-k}$ and hence exhibits optimal rate of convergence. On the other hand, the h-version shows locking of order $h^{-2}$, and is uniformly robust with order $h^{p-2}$ for $p>2$ which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience. If mixed method is used, both the h-version and the p-version are free from locking. Furthermore, the mixed method even converges uniformly with an optimal rate for the stress.