University of Illinois Urbana-Champaign

Conservation and efficiency in least squares finite element methods

Lai, James

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Lai_James.pdf

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https://hdl.handle.net/2142/34237

Description

Title
Conservation and efficiency in least squares finite element methods
Author(s)
Lai, James
Issue Date
2012-09-18T21:07:20Z
Director of Research (if dissertation) or Advisor (if thesis)
Olson, Luke N.
Doctoral Committee Chair(s)
Olson, Luke N.
Committee Member(s)
  • Bochev, Pavel B.
  • Heath, Michael T.
  • Gropp, William D.
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2012-09-18T21:07:20Z
Keyword(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.
Graduation Semester
2012-08
Permalink
http://hdl.handle.net/2142/34237
Copyright and License Information
Copyright 2012 James Lai

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