Summary: INTERIOR ESTIMATES FOR A LOW ORDER FINITE ELEMENT METHOD
FOR THE REISSNER­MINDLIN PLATE MODEL*
DOUGLAS N. ARNOLD and XIAOBO LIU
Abstract. Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner­
Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded
above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the
global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the
Arnold­Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even
though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support
our conclusion.
Key words. Reissner­Mindlin plate, boundary layer, mixed finite element, interior error estimate
AMS(MOS) subject classifications (1991 revision). 65N30, 73N10
1. Introduction
The Reissner­Mindlin plate model describes deformation of a plate with small to moderate thickness
subject to a transverse load. The finite element method for this model has been studied extensively (cf. [6],
[9], [12], and references therein) and it has been known for a long time that a direct application of standard
finite element methods usually leads to unreasonably small solution, as the plate thickness approaches zero.
This is usually called the "locking" phenomenon of the finite element method for the Reissner­Mindlin plate.
Another difficulty in approximating the Reissner­Mindlin plate equations is that the solution possesses
boundary layers. The structure of the dependence of the solution on the plate thickness was analyzed in

Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota

Collections: Mathematics