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Title:Conservation and efficiency in least squares finite element methods
Author(s):Lai, James
Director of Research:Olson, Luke N.
Doctoral Committee Chair(s):Olson, Luke N.
Doctoral Committee Member(s):Bochev, Pavel B.; Heath, Michael T.; Gropp, William D.
Department / Program:Computer Science
Discipline:Computer Science
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):finite elements
least-squares finite element methods
multigrid
high-order
discontinuous least-squares
Navier-Stokes
Stokes
edge elements
H(curl) multigrid
Abstract:Two of the main aspects in the numerical solution of partial differential equations include accurate discretizations and efficient solutions of the algebraic equations. With respect to discretizations, conservation is often sought after. However, least-squares finite element methods are known to be not mass conserving when solving fluid flow problems. In this dissertation we develop mass conservative least-squares finite element methods for the Stokes and Navier-Stokes equations through the use of discontinuous finite element spaces. We formulate two divergence free formulations using both a discontinuous stream-function and a locally divergence free basis and we present a thorough numerical study of both methods. This dissertation is also concerned with the efficient solution of algebraic equations via multigrid methods. Specifically, we formulate multigrid methods for high-order H(curl) conforming finite elements. Such elements are often used in mimetic discretizations of Maxwell's equations often solved in electromagnetic applications. Efficient multigrid methods for high-order H^1 conforming finite elements and also for the lowest-order H(curl) basis have been extensively studied in recent research. We draw upon elements of both algorithms to formulate multigrid methods for high-order H(curl) finite elements for hierarchical and interpolatory type.
Issue Date:2012-09-18
URI:http://hdl.handle.net/2142/34237
Rights Information:Copyright 2012 James Lai
Date Available in IDEALS:2012-09-18
Date Deposited:2012-08


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