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Mathematics of Computation

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On the convergence of a finite element method for a nonlinear hyperbolic conservation law

Authors: Claes Johnson and Anders Szepessy
Journal: Math. Comp. 49 (1987), 427-444
MSC: Primary 65M10; Secondary 65M60
MathSciNet review: 906180
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Abstract: We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers’ equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function u, then u is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers’ equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments.

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Keywords: Finite element method, convergence, nonlinear hyperbolic conservation law
Article copyright: © Copyright 1987 American Mathematical Society