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On the convergence of higher-order finite element methods to weak solutions

Author(s)
Couchman, Benjamin Luke Streatfield
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Massachusetts Institute of Technology. Department of Aeronautics and Astronautics.
Advisor
David L. Darmofal.
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MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission. http://dspace.mit.edu/handle/1721.1/7582
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Abstract
The ability to handle discontinuities appropriately is essential when solving nonlinear hyperbolic partial differential equations (PDEs). Discrete solutions to the PDE must converge to weak solutions in order for the discontinuity propagation speed to be correct. As shown by the Lax-Wendroff theorem, one method to guarantee that convergence, if it occurs, will be to a weak solution is to use a discretely conservative scheme. However, discrete conservation is not a strict requirement for convergence to a weak solution. This suggests a hierarchy of discretizations, where discretely conservative schemes are a subset of the larger class of methods that converge to the weak solution. We show here that a range of finite element methods converge to the weak solution without using discrete conservation arguments. The effect of using quadrature rules to approximate integrals is also considered. In addition, we show that solutions using non-conservation working variables also converge to weak solutions.
Description
Thesis: S.M., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2018.
 
Cataloged from PDF version of thesis.
 
Includes bibliographical references (pages 77-79).
 
Date issued
2018
URI
http://hdl.handle.net/1721.1/115685
Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Publisher
Massachusetts Institute of Technology
Keywords
Aeronautics and Astronautics.

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