A numerical technique is presented for determining a simple turning point in a branch of solutions of an algebraic system of equations depending on a scalar parameter. Results are given from testing the method on discrete versions of several mildly nonlinear boundary value problems to determine turning points in the positive solution branch.

A Method for the Numerical Determination of Bifurcation States of Nonlinear Systems of Equations

We present a new method for the numerical solution of bifurcation problems for ordinary differential equations. It is based on a modification of the classical Ljapunov-Schmidt-theory. We transform the problem of determining the nontrivial branch bifurcating from the trivial solution into the problem of solving regular nonlinear boundary value problems, which can be treated numerically by standard methods (multiple shooting, difference methods).

Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen

Finite-difference methods using the five-point discrete Laplacian and suit- able boundary modifications for approximating (1) - Au = Xf(x, u) in a plane domain D, u = 0 on its boundary are considered. It is shown that if (1) has an isolated solution, u, then the discrete problem has a solution, Uh, for which Uh - u = 0(h2). If the discrete problem has solutions, U,,, such that I Uh I X*, then the discrete problem has an analogous critical value X*? and, under suitable conditions, X* -* = 0(h 4/3-), e > 0. Computed results for the case j(x, u) = eu and D the unit square are given. Introduction. For some nonlinear elliptic boundary-value problems, the exist- ence and number of solutions depends strongly on the values of parameters in the problem. In this paper, we present some affirmative answers to the question of whether discrete versions of the problems have the same number of solutions for fixed param- eter values; and we establish some asymptotic error estimates for these cases. The problems to be considered are perhaps the simplest ones in which this phenomenon appears nontrivially, the problems of finding solutions, u(x; X), to the mildly nonlinear problem for a bounded domain D of the plane (0.1) -A u(x) = Xf(X, u(X)), x C D, (0.2) u(x) = 0, x E AD. In particular, we are interested in nonlinearities, f(x, u), of the type occurring in certain nonlinear steady-state diffusion problems. For these problems, positive solu- tions of (0.1), (0.2) are desired and, typically, these exist for certain ranges of X, 0 < X < X*, where X* and the multiplicity of positive solutions are difficult to predict

/pdf/existence-and-error-estimates-for-solutions-of-a-discrete-1o49sxudqq.pdf

Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems

The approximation of solutions of nonlinear elliptic boundary value problems having several solutions

A numerical method is developed for approximating the exact solution of an operator equation on a certain finite grid to within a desired tolerance. The method incorporates discretizations which admit asymptotic expansions of the error, mesh refinement strategies and discrete Newton methods. An algorithm is given in which essentially the largest adequate mesh size is used. A homotopy method for obtaining good starting values for a Newton-type method applied to a coarse grid discretization and the connection that our approach has with multigrid methods are discussed. Numerical examples for two-point boundary value problems and elliptic boundary value problems are given.

Discrete correction methods for operator equations

Finite Difference Methods for Mildly Nonlinear Eigenvalue Problems

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Finite Difference Methods for Mildly Nonlinear Eigenvalue Problems