Algorithms Defined by Nash Iteration: Some Implementations via Multilevel Collocation and Smoothing

We describe the general algorithms of Nash iteration in numerical analysis. We make a particular choice of algorithm involving multilevel collocation and smoothing. Our test case is that of a linear differential equation, although the theory allows for the approximate solution of nonlinear differential equations. We describe the general situation completely, and employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps. We had earlier shown how these ideas apply to scattered data approximation, but in this work we are interested in the application of the ideas to the numerical solution of differential equations. We make use of approximate smoothers, involving the solution of evolution equations with calibrated time steps. Acknowledgments: The second author is supported by the National Science Foundation under grant DMS-9870420. The third author is supported by the National Science Foundation under grant DMS-9704458. We thank the referee for a number of helpful comments. ∗Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616. email:[email protected] †Department of Mathematics and Comp. Sc. , Kent State University, Kent, OH 44242. email: [email protected] ‡Department of Mathematics, Northwestern University, Evanston, IL 60208. email: [email protected]; corresponding author.