Conservation and efficiency in least squares finite element methods

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2012

Author dc:creator

• Lai, James

Contributors dc:contributor

• Olson, Luke N.
• Bochev, Pavel B.
• Heath, Michael T.
• Gropp, William D.

• finite elements
• least-squares finite element methods
• multigrid
• high-order
• discontinuous least-squares
• Navier-Stokes
• Stokes
• edge elements
• H(curl) multigrid

Statement dc:rights

• Copyright 2012 James Lai

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/34237