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

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906180-5

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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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906180-5

Keywords:
Finite element method,
convergence,
nonlinear hyperbolic conservation law

Article copyright:
© Copyright 1987
American Mathematical Society