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 Title: High-order numerical methods for layer potential evaluation Author(s): Wala, Mateusz Michal Director of Research: Klöckner, Andreas Doctoral Committee Chair(s): Klöckner, Andreas Doctoral Committee Member(s): Olson, Luke; Fischer, Paul; Greengard, Leslie Department / Program: Computer Science Discipline: Computer Science Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): integral equations layer potentials fast multipole method singular integrals quadrature by expansion conformal mapping time integration Abstract: This thesis addresses a number of obstacles in the practical realization of integral equation methods for boundary value problems of elliptic partial differential equations. Layer potentials play a central role in the integral equation formulation of boundary value problems. However, the numerical evaluation of layer potentials presents significant practical challenges associated with issues of singular/near-singular quadrature and efficient scaling. Quadrature by Expansion (QBX) is a quadrature method that addresses many of the issues encountered in the evaluation of layer potentials, by providing a high-order and kernel/dimension independent quadrature scheme that is also acceleration-compatible, i.e., able to be integrated with a fast summation scheme to reduce the cost of $O(N^2)$ pairwise interactions to $O(N)$. The focus of the first part of this thesis is an acceleration scheme for QBX based on the Fast Multipole Method (FMM) that provides both high performance and strong accuracy guarantees for mathematical control over the error introduced by acceleration. The scheme extends to both two and three dimensions and Laplace and Helmholtz kernels. The contribution in this thesis also includes a geometry processing framework to prepare arbitrary smooth geometries for accurate quadrature with QBX, a comprehensive complexity and error analysis of the algorithm, and a cost model and study of key optimizations. This thesis also considers the application of layer potentials to the problem of numerical conformal mapping. It is shown how there is a strong connection between the double-layer potential and the Faber/Faber-Laurent polynomials, which are polynomials closely related to the series expansion of the Riemann map. From this, a scheme for computing the Riemann map of a piecewise smooth Jordan domain is devised. This latter work also provides an insight into the convergence behavior of QBX. Finally, this thesis describes a software abstraction for the design and implementation of time integration algorithms for the solution of initial value problems. This abstraction, Dagrt, has two aims. The first is a epresentation that decouples the mathematical specification of a time integration algorithm from its realization in a particular programming model, and the second is to give the user control over implementation details. We demonstrate the capabilities of this abstraction by presenting Leap, a collection of pre-written time integration algorithm specifications that includes complex multistep and multistage methods. Issue Date: 2019-12-02 Type: Text URI: http://hdl.handle.net/2142/106232 Rights Information: Copyright 2019 Mateusz Wala Date Available in IDEALS: 2020-03-02 Date Deposited: 2019-12
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