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On the Convergence of a Finite Element Method for a Nonlinear Hyperbolic Conservation Law
Claes Johnson and Anders Szepessy
Mathematics of Computation
Vol. 49, No. 180 (Oct., 1987), pp. 427-444
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2008320
Page Count: 18
You can always find the topics here!Topics: Entropy, Conservation laws, Finite element method, Mathematical functions, Mathematical problems, Mallets, Topological compactness, Shock tests, Mathematics, Dental calculus
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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.
Mathematics of Computation © 1987 American Mathematical Society