Document Type

Dissertation

Date of Award

Fall 12-31-2018

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Catalin C. Turc

Second Advisor

Yassine Boubendir

Third Advisor

Cyrill B. Muratov

Fourth Advisor

Peter G. Petropoulos

Fifth Advisor

Francisco-Javier Sayas

Abstract

This presents a Schur complement Domain Decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods,(1) the ensemble of scatterers is enclosed in a domain bounded by an artificial boundary, (2) this domain is subdivided into a collection of nonoverlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers, and (3) the solutions of the subproblems are connected via Robin boundary conditions matching on the common interfaces between subdomains. Subdomain Robin-to-Robin maps are used to recast the DD problem as a sparse linear system whose unknown consists of Robin data on the interfaces between sub domains-two unknowns per interface. The Robin-to-Robin maps are computed in terms of well-conditioned boundary integral operators. Unlike classical DD, the Domain Decomposition problem is not reformulated in the form of a fixed point iteration, but rather solved as a linear system through Gaussian elimination of the unknowns corresponding to inner interfaces between subdomains via Schur complements. Once all the unknowns corresponding to inner subdomains interfaces have been eliminated, a much smaller linear system involving solely the unknowns on the inner and outer artificial boundary is solved. The last section of this thesis offers numerical evidence that this Schur complement DD algorithm can produce accurate solutions for very large multiple scattering problems that are out of reach for other existing approaches.

Included in

Mathematics Commons

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