High Order Boundary Perturbation Methods for Boundary Value and Free Boundary Problems

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

DMS 0072462 ABSTRACT. The subject of this proposal is the development (both numerical and analytical) of a new class of perturbative methods for estimating solutions of boundary value problems (BVP) and free boundary problems (FBP) arising in mathematical physics. Oscar Bruno and Fernando Reitich recently proposed a new class of perturbation methods for approximating solutions of BVP and FBP which are based on the ideas of classical perturbation theory applied to domains which are small deviations from exactly solvable geometries. These methods are interesting because they are fast, easy to implement, and translate into three and higher dimensions without major modification. Of course, such schemes are limited by the extent of their domain of convergence, which may be quite small, and the fact that in many BVP and FBP of interest, the domain is a large perturbation of a simple geometry. This obstacle has been effectively overcome in recent work of Bruno & Reitich in the setting of acoustic and electromagnetic scattering via the introduction of analytic continuation techniques, in particular the use of Pade approximants. A second challenge faced by the current class of perturbative methods is that they suffer from problems of numerical ill-conditioning due to subtle cancellations which take place in their evaluation. This drawback has been overcome by the PI & Reitich, for the problem of computing Dirichlet-Neumann operators (DNO) for Laplace's equation, via a straightforward change of variables which simply flattens the domain. The PI proposes to extend the above results by developing a general purpose perturbative method for solving BVP and FBP which incorporates both analytic continuation and domain flattening techniques. To date, the techniques have only been applied independently to BVP. The first objective is to first implement them simultaneously for the BVP of computing DNO (for Laplace's equation), and then extend these methods to the case of a genuine FBP (modeling the motion of the interface of an ideal fluid) dimensions. An investigation of the effects of bottom topography and multiple fluid layers on the methods will follow, and considerations will be made of other classical FBP (e.g. Hele-Shaw flows, Stefan problems, etc.) whose geometries will pose their own challenges. Subsequently a thorough re-investigation of the problems of electromagnetic and acoustic scattering will be completed with domain flattening techniques implemented to overcome numerical ill-conditioning problems. Finally, the problem of implementing transparent boundary conditions in scattering problems via DNO will be considered. Many important scientific problems are defined on complicated domains that may or may not evolve in time. These problems, such as the scattering of electromagnetic radiation from a rough surface or the evolution of surface waves on a fluid, pose severe theoretical and computational difficulties for applied mathematicians and engineers. When the domain of the problem is simple (rectangular, circular, etc.) the problem can usually be solved explicitly by classical methods. One approach to the estimation of more general problems is to first consider domains which are small deviations from simple geometries. Many approaches along these lines have been proposed, but unless great care is taken, they can result in approximations that actually degrade as the approximation is refined. A new technique, developed by the PI & Fernando Reitich, avoids such difficulties and provides an exciting new method for the estimation of problems on complicated geometries. However, challenges still remain and these are the subject of this proposal. One challenge is to extend our new methods for small domain deviations to problems which are large deviations from a simple geometry. Another challenge is the fast and efficient implementation of these new methods on high performance serial and parallel computers.