New Approaches to Bifurcation Analysis of Nonlinear Elliptic PDEs

Dr. Mark Friedman

Department of Mathematical Sciences
University of Alabama in Huntsville

January 29, 1999

Abstract

Nonlinear elliptic PDEs are the basis for many scientific and engineering problems. In these problems it is crucial to understand the qualitative dependence of the solution on the problem parameters. The area of applied mathematics which deals with these issues is numerical bifurcation theory.

the nonlinear elliptic PDEs are discretized by the Multiquadric Radial Basis Function(MQ) Method, a highly accurate meshless collocation method, with global basis functions. The resulting nonlinear systems, though not sparse, are of dramatically smaller size than those resulting from a standard; e.g., finite difference or finite element discretization. These nonlinear systems can be efficiently continued by existing continuation software, such as AUTO97 or CONTENT98.

We also formulate a new approach for reliable detection of bifurcations in large systems. The approach is based on a new algorithm for continuation of invariant subspaces of a parameter dependent matrix.

The presentation is elementary and is accessible for those familiar with calculus and linear algebra.