On numerical approximation in bifurcation theory. (English) Zbl 0687.65057

Crouzeix, Michel; Rappaz, Jacques

Recherches en Mathématiques Appliquées, 13. Paris etc.: Masson; Berlin etc.: Springer-Verlag. ix, 164 p. (1989).

Consider a parametrized equation \((1)\quad F(\lambda,u)=0,\) where \(F: {\mathbb{R}}\times V\to Z\) and V, Z are real Banach spaces. Typically, for the computation of solution branches of (1) finite-dimensional approximations \((2)\quad F_ h(\lambda,u)=0\) of (1) have to be introduced. This monograph presents an analysis of the convergence and of error estimates for the solutions of (2), when u is a regular solution, a turning point or a simple bifurcation point of (1). After several introductory examples, some results about the approximation error of linear elliptic problems are given, and in Chapter 3 the main results about the approximation of regular solution branches of (1) are developed.
These results are applied in Chapter 4 and 5 to the case of a simple limit point and a simple bifurcation point on the trivial branch, respectively. Then, in Chapter 6 a form of the Lyapunov-Schmidt procedure is used to handle simple bifurcations from general branches and Chapter 7 gives an outlook to related problems. The results in this monograph are in essence generalized versions of the earlier results of F. Brezzi, J. Rappaz and P. A. Raviart [Numer. Math. 36, 1-25 (1980; Zbl 0488.65021), ibid. 37, 1-28 (1981; Zbl 0525.65036), ibid. 38, 1-30 (1981; Zbl 0525.65037)] and J. Descloux and J. Rappaz [RAIRO, Anal. Numer. 16, 319-349 (1982; Zbl 0505.65016)]. There is no consideration of other approaches to this approximation problem, as, for example, those of J. Fink and the reviewer [SIAM J. Numer. Anal. 20, 732-746 (1983; Zbl 0525.65041) and Numer. Math. 45, 323-343 (1984; Zbl 0555.65034)].

Reviewer: W.C.Rheinboldt

MSC:

65J15	Numerical solutions to equations with nonlinear operators
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
65H17	Numerical solution of nonlinear eigenvalue and eigenvector problems
47J25	Iterative procedures involving nonlinear operators
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

Banach spaces; solution branches; monograph; convergence; error estimates; turning point; bifurcation point; regular solution branches; Lyapunov-Schmidt procedure

Citations:

Zbl 0488.65021; Zbl 0525.65036; Zbl 0525.65037; Zbl 0505.65016; Zbl 0525.65041; Zbl 0555.65034