Date of Award

Summer 2017

Document Type

Dissertation

Degree Name

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

Department

Mathematical Sciences

First Advisor

Catalin C. Turc

Second Advisor

Michael Siegel

Third Advisor

Eliza Zoi-Heleni Michalopoulou

Fourth Advisor

Peter G. Petropoulos

Fifth Advisor

Francisco-Javier Sayas

Abstract

In this dissertation, an in-depth comparison between boundary integral equation solvers and Domain Decomposition Methods (DDM) for frequency domain Helmholtz transmission problems in composite two-dimensional media is presented. Composite media are characterized by piece-wise constant material properties (i.e., index of refraction) and thus, they exhibit interfaces of material discontinuity and multiple junctions. Whenever possible to use, boundary integral methods for solution of Helmholtz boundary value problems are computationally advantageous. Indeed, in addition to the dimensional reduction and straightforward enforcement of the radiation conditions that these methods enjoy, they do not suffer from the pollution effect present in volumetric discretization. The reformulation of Helmholtz transmission problems in composite media in terms of boundary integral equations via multi-traces constitutes one of the recent success stories in the boundary integral equation community. Multi-trace formulations (MTF) incorporate local Dirichlet and Neumann traces on subdomains within Green’s identities and use restriction and extension by zero operators to enforce the intradomain continuity of the fields and fluxes. Through usage of subdomain Calderon projectors, the transmission problem is cast into a linear system form whose unknowns are local Dirichlet and Neumann traces (two such traces per interface of material discontinuity) and whose operator matrix consists of diagonal block boundary integral operators associated with the subdomains and extension/projections off diagonal blocks. This particular form of the matrix operator associated with MTF is amenable to operator preconditioning via Calderon projectors.

Included in

Mathematics Commons

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