On the Convergence of a Finite Element Method for a Nonlinear Hyperbolic Conservation Law
- 1 October 1987
- journal article
- Published by JSTOR in Mathematics of Computation
- Vol. 49 (180) , 427-444
- https://doi.org/10.2307/2008320
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.Keywords
This publication has 11 references indexed in Scilit:
- A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systemsComputer Methods in Applied Mechanics and Engineering, 1986
- A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systemsComputer Methods in Applied Mechanics and Engineering, 1986
- BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than oneCommunications in Mathematical Physics, 1986
- Streamline Diffusion Methods for the Incompressible Euler and Navier-Stokes EquationsMathematics of Computation, 1986
- A new finite element formulation for computational fluid dynamics: II. Beyond SUPGComputer Methods in Applied Mechanics and Engineering, 1986
- A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamicsComputer Methods in Applied Mechanics and Engineering, 1986
- Finite element methods for linear hyperbolic problemsComputer Methods in Applied Mechanics and Engineering, 1984
- Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible euler equationsComputer Methods in Applied Mechanics and Engineering, 1984
- Convergence of approximate solutions to conservation lawsArchive for Rational Mechanics and Analysis, 1983
- On the symmetric form of systems of conservation laws with entropyJournal of Computational Physics, 1983