Investigation of behaviour of approximate solution of a nonlinear boundary value problem in the instability area

Abstract

A numerical solution of the nonlinear boundary-value problem is obtained by means of the finite
elements method. Variational relation of a problem of stability analysis in the vicinity of any
solution of the nonlinear problem is formulated. This problem is discretized by means of the same
approximations of finite elements methods, as the solution of the nonlinear problem. The
eigenvalue matrix partial problems by a method of iterations in a subspace are reduced to a
sequence of full matrix problems of small dimension. These problems are transformed to threediagonal
ones by Householder method and solved by QL-algorithm. The critical value and
corresponding forms of loss of stability in the vicinity of the initial and the final states are
analysed.