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On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws


Authors: Claes Johnson, Anders Szepessy and Peter Hansbo
Journal: Math. Comp. 54 (1990), 107-129
MSC: Primary 65M60; Secondary 35L65, 76L05
DOI: https://doi.org/10.1090/S0025-5718-1990-0995210-0
MathSciNet review: 995210
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Abstract: We extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock-capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the mesh size. With this term present, we prove a maximum norm bound for finite element solutions of Burgers' equation and thus complete an earlier convergence proof for this equation. We further prove, using entropy variables, that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality associated with the entropy variables. Results of some numerical experiments for the time-dependent compressible Euler equations in two dimensions are also reported.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-0995210-0
Keywords: Finite element method, conservation laws, convergence streamline diffusion, shock-capturing, entropy variables
Article copyright: © Copyright 1990 American Mathematical Society

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