Date of Award
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Peter G. Petropoulos
David James Horntrop
Young Ju Lee
The purpose of this thesis is to formulate and investigate new iterative methods for the solution of scattering problems based on the domain decomposition approach. This work is divided into three parts. In the first part, a new domain decomposition method for the perfectly matched layer system of equations is presented. Analysis of a simple model problem shows that the convergence of the new algorithm is guaranteed provided that a non-local, square-root transmission operator is used. For efficiency, in practical simulations such operators need to be localized. Current, state of the art domain decomposition algorithms use the localization technique based on rational approximation of the symbol of the transmission operator. However, the original formulation of the procedure assumed decompositions that contain no cross-points and consequently could not be used in the cross-point algorithm. In the context of the perfectly matched layer problem, we adapt the cross-point technique and combined with the rational approximation of the square root transmission operator to yield an effective algorithm. Furthermore, to reduce Krylov subspace iterations, we present a new, adequate and efficient preconditioner for the perfectly matched layer problem. The new, zero frequency limit preconditioner shows great reduction in the required number of iterations while being extremely easy to construct.
In the second part of the thesis, a new domain decomposition algorithm is considered. From theoretical point of view, its formulation guarantees well-posedness of local problems. Its practicality on the other hand is evident from its efficiency and ease of implementations as compared with other, state of the art domain decomposition approaches. Moreover, the method exhibits robustness with respect to the problem frequency and is suitable for large scale simulations on a parallel computer.
Finally, the third part of the thesis presents an extensible, object oriented architecture that supports development of parallel domain decomposition algorithms where local problems are solved by the finite element method. The design hides mesh implementation details and is capable of supporting various families of finite elements together with quadrature formulas of suitable degree of precision.
Midura, Dawid, "Efficient domain decomposition algorithms for the solution of the helmholtz equation" (2014). Dissertations. 1615.