Conservation and efficiency in least squares finite element methods

Lai, James

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

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

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

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

Copyright and License Information

Copyright 2012 James Lai

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