"Finite Elements. Theory, Fast Solvers
and Applications in Solid Mechanics".
Cambridge University Press 2001 (April)
ISBN: 0 521 01195 7
We list some new material from the 2001 edition.
The introduction of finite element spaces in Chapter II, Section 5
is now focused such that all the ingredients of the formal
definition at the end of this section are well motivated.
There are more estimates for the interpolation of Clement type.
The general considerations of saddle point problems in
Chapter III are augmented. The direct and converse theorems that are
related to Fortin interpolation are presented now under a common
aspect. Mixed methods are often connected with a softening of
the energy functional that is wanted in some applications for good
reasons. A different but equivalent variational formulation
that has become popular in solid mechanics, is easily understood.
In Chapter IV only the (short) standard proof of the Kantorowitch inequality
has been replaced by a shorter one. The monotonicity of the
function t+1/t (t>1) and Young's inequality are sufficient.
The multigrid theory requires less regularity assumptions
if convergence with respect to the energy norm is considered.
A quick introduction into that theory is now included,
and multigrid algorithms are also considered in the framework
of space decompositions.
Finite element computations in solid mechanics require often
appropriate elements in order to avoid an effect called "locking"
by engineers. From the mathematical point of view we have problems
with a small parameter. Methods for
treating nearly incompressible material serve as a model for
positive results while negative results are easily described
for a more general framework.
Solutions of selected problems
are provided.
Hints to some updates and to misprints (if necessary) will be
given in due time on my web page.