On the convergence of higher-order finite element methods to weak solutions
Author(s)Couchman, Benjamin Luke Streatfield
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics.
David L. Darmofal.
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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.
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).
DepartmentMassachusetts Institute of Technology. Department of Aeronautics and Astronautics.; Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Massachusetts Institute of Technology
Aeronautics and Astronautics.