Summary: Numer. Math. 37, 405-421 (1981) Numerische
Discretization by Finite Elements
of a Model Parameter Dependent Problem
Douglas N. Arnold
Department of Mathematicsand Institute for PhysicalScienceand Technology.
Universityof Maryland,ColIegePark, MaryIand 20742, USA
Abstract. The discretization by finite elements of a model variational prob-
lem for a clamped loaded beam is studied with emphasis on the effect of
the beam thickness, which appears as a parameter in the problem, on the
accuracy. It is shown that the approximation achieved by a standard finite
element method degenerates for thin beams. In contrast a large family of
mixed finite element methods are shown to yield quasioptimal approxima-
tion independent of the thickness parameter. The most useful of these
methods may be realized by replacing the integrals appearing in the stiff-
ness matrix of the standard method by Gauss quadratures.
Subject Classifications: AMS(MOS): 65N30; CR: 5.17.
In this paper we examine the finite element discretization of a model vari-