Finite element analysis for parametrized nonlinear equations around turning points

Nonlinear equations with parameters are called parametrized nonlinear equations. In this paper, a priori error estimates of finite element solutions of parametrized nonlinear elliptic equations on branches around turning points are considered. Existence of a finite element solution branch is shown under suitable conditions on an exact solution branch around a turning point. Also, some error estimates of distance between exact and finite element solution branches are given. It is shown that error of a parameter is much smaller than that of functions. Approximation of nondegenerate turning points is also considered. We show that if a turning point is nondegenerate, there exists a locally unique finite element nondegenerate turning point. At a nondegenerate turning point an elaborate error estimate of the parameter is proved.