In this review we discuss bifurcation theory in a Banach space setting using
the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation
points. The numerical analysis of bifurcation problems is discussed
and the convergence theory for several important bifurcations is described for
both projection and finite difference methods. These results are used to provide
a convergence theory for the mixed finite element method applied to the
steady incompressible Navier–Stokes equations. Numerical methods for the
calculation of several common bifurcations are described and the performance
of these methods is illustrated by application to several problems in fluid mechanics.
A detailed description of the Taylor–Couette problem is given, and
extensive numerical and experimental results are provided for comparison and
discussion.
DIFFUSION AND BIFURCATION PROBLEMS IN SINGULARLY PERTURBED DOMAINS