The approximation of generalized turning points by projection methods with superconvergence to the critical parameter (Q1078981)

The authors consider mildly nonlinear equations (1) \(F(z,\lambda)=x+f(x,\beta,\lambda)=0\) where \(z=(x,\beta)\) and F is a \(C^ r\)-map from an open set in \(Z=X\times R^{p-1}\) into the Banach space X. The range of \(F_ z\) is assumed to have co-dimension at most one. For the computation of singular solutions, where this co-dimension is exactly one, augmented systems of the form (2) \(F(z,\lambda)=0\), \(g_ i(z,\lambda)=0\), \(i=1,...,p\), are constructed as in the authors' earlier paper in [SIAM J. Numer. Anal. 21, 176-185 (1984; Zbl 0536.65031)]. Then the error is analyzed when the characterizing equations (2) are discretized by some general projection scheme. These results are extensions of those of \textit{F. Brezzi}, \textit{J. Rappaz}, \textit{P. Raviart} [Numer. Math. 36, 1-25 (1980; Zbl 0488.65021), 37, 1-28 (1981; Zbl 0525.65036) and 38, 1-30 (1981; Zbl 0525.65037)] and are closely related to those of \textit{J. P. Fink} and the reviewer [ibid. 45, 323-343 (1984; Zbl 0555.65034)]. A procedure is developed for the computation of the singular points. Moreover, for certain two point boundary value problems collocation at Gauss points is shown to achieve superconvergence.