Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 27

Chapter II

Conforming Finite Elements

The mathematical treatment of the finite element method is based on the varia-
tional formulation of elliptic differential equations. Solutions of the most important
differential equations can be characterized by minimal properties, and the corre-
sponding variational problems have solutions in certain function spaces called
Sobolev spaces. The numerical treatment involves minimization in appropriate
finite-dimensional linear subspaces. A suitable choice for these subspaces, both
from a practical and from a theoretical point of view, are the so-called finite ele-
ment spaces .
For linear differential equations, it suffices to work with Hilbert space meth-
ods. In this framework, we immediately get the existence of so-called weak solu-
tions . Regularity results, to the extent they are needed for the finite element theory,
will be presented without proof.
This chapter contains a theory of the simple methods which suffice for the
treatment of scalar elliptic differential equations of second order. The aim of this
chapter are the error estimates in §7 for the finite element solutions. They refer
to the L 2 -norm and to the Sobolev norm

· 1 . Some of the more general results
presented here will also be used later in our discussion in Chapter III of other
elliptic problems whose treatment requires additional techniques.
The paper of Courant [1943] is generally considered to be the first mathemat-
ical contribution to a finite element theory, although a paper of Schellbach [1851]
had appeared already a century earlier. If we don't take too narrow a view, finite
elements also appear in some work of Euler. The method first became popular at
the end of the sixties, when engineers independently developed and named the
method. The long survey article of Babuska and Aziz [1972] laid a broad founda-
tion for many of the deeper functional analytic tools, and the first textbook on the
subject was written by Strang and Fix [1973].
Independently, the method of finite elements became an established technique
in engineering sciences for computations in structural mechanics. The develop-
ments there began around 1956, e.g., with the paper of Turner, Clough, Martin,
and Topp [1956] who also created the name finite elements and the paper by Ar-
gyris [1957]. The topic by Zienkiewicz [1971] also had great impact. An interesting
review of the history was presented by Oden [1991].