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Title:High-order hybrid methods using Green's functions and finite elements
Author(s):Beams, Natalie N
Director of Research:Olson, Luke
Doctoral Committee Chair(s):Fischer, Paul
Doctoral Committee Member(s):Kloeckner, Andreas; West, Matthew
Department / Program:Mechanical Sci & Engineering
Discipline:Theoretical & Applied Mechans
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Finite element
Integral equation
Green's functions
Abstract:Green's functions provide an elegant mathematical framework for linear partial differential boundary value problems by directly defining solutions in integral form. For a Green's function that satisfies the desired boundary conditions, this form is a convolution of the Green's function with the right hand side. However, straightforward numerical computation of the convolution generally results in a prohibitively costly algorithm. This dissertation focuses on combining Green's function-based computations with finite element (FE) methods to alleviate their computational challenges while enhancing the representation capability of standard FE solutions on structured or semi-structured meshes. Two main areas of application are considered: a new finite-element-based method for N-body calculations and a new high-order method for elliptic interface problems on non-conforming volume meshes. First, the FE-based particle-particle--particle-mesh (FE-P3M) method for N-body problems is introduced. P3M methods are designed to compute quantities (e.g. potentials or forces) that are sums of discrete Green's functions and represent the effects of N interacting bodies. To avoid an N^2 operation when the quantity is desired at all body locations (as is often the case), P3M methods decompose the potential into short-range interactions between near neighbors and long-range interactions that are smooth and readily solved by mesh-based numerical methods. Instead of taking the traditional "Ewald-based" approach of using Gaussian screen functions to accomplish this decomposition, the FE-P3M method builds specially designed polynomial screens on an introduced finite element mesh. Due to this form of the screens, the long-range component of the potential is accurately resolved with a finite element method. When compared to classic P3M methods, which rely on the fast Fourier transform (FFT), the FE-P3M method allows for more flexible boundary conditions and possibly much lower communication costs in a parallel implementation. The method is described in detail, its cost and memory requirements are discussed, and its accuracy is demonstrated. The remaining chapters consider Green's functions in the context of layer potential solutions to homogeneous boundary value problems and the resulting integral equations (IEs) formulated on the boundary of the domain. A high-order coupling of finite elements and integral equations is presented for the solution of elliptic interface problems on embedded domains. This FE-IE method requires no special basis functions nor modifications to handle homogeneous or non-homogeneous jump conditions, and retains high-order convergence close to the embedded interface. The interface or embedded boundary conditions are enforced directly at the discretization points of the embedded surface mesh. High-order numerical convergence is demonstrated for both interior and exterior embedded domain problems, with a novel splitting defined and analyzed for the latter. Theoretical error convergence, existence and uniqueness of the decomposition, and numerical solution considerations are discussed. The new FE-IE interface method is then shown to be built from interior and exterior FE-IE subproblems. Finally, an extension to Stokes flow around embedded objects in two and three dimensions is presented.
Issue Date:2017-07-13
Rights Information:Copyright 2017 Natalie Beams
Date Available in IDEALS:2017-09-29
Date Deposited:2017-08

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