Mathematical Sciences: Development and Applications of Iterative Methods for Solving Nonlinear Elliptic Eigenvalue Problems

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With this award the principal investigator will continue to develop numerical methods based upon iterative techniques for the approximate solution of a class of nonlinear elliptic eigenvalue problems that arises in physics and engineering. What sets this class of problems apart and makes it interesting is the requirement that the solutions to be approximated must satisfy an additional integral constraint. Most numerical schemes in current use approximate solutions of elliptic equations that satisfy conditions on the boundary of the domain, while in this grant the principal investigator has to modify existing schemes or invent new ones in order to assure satisfaction of the integral constraint. Such eigenvalue problems arise in the theory of nonlinear waves and there is a genuine need for reliable numerical methods to generate accurate approximate solutions. In many mathematical models of physical phenomena the process under consideration is realized as the solution of a differential equation that satisfies certain additional conditions. For example, one may know the state of the process at some initial time or along the boundary of the domain. In these cases there exist reliable numerical methods that provide accurate approximate solutions with the aid of a computer. In this grant the principal investigator will study processes that satisfy an integral constraint, instead of the more familiar initial or boundary conditions alluded to above, and so she will have to develop new numerical methods to find the solution.