On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws
HTML articles powered by AMS MathViewer
- by Claes Johnson, Anders Szepessy and Peter Hansbo PDF
- Math. Comp. 54 (1990), 107-129 Request permission
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.References
- Kenneth Eriksson and Claes Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988), no. 182, 361–383. MR 929542, DOI 10.1090/S0025-5718-1988-0929542-X P. Hansbo, Finite element procedures for conduction and convection problems, Publication 86:7, Dept. of Structural Mechanics, Chalmers Univ. of Technology, S-412 96 Göteborg, 1986. —, Streamline diffusion methods and adaptive procedures in finite element methods, Thesis, Dept. of Structural Mechanics, Chalmers Univ. of Technology, 1989.
- Amiram Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys. 49 (1983), no. 1, 151–164. MR 694161, DOI 10.1016/0021-9991(83)90118-3
- Alexander N. Brooks and Thomas J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), no. 1-3, 199–259. FENOMECH ”81, Part I (Stuttgart, 1981). MR 679322, DOI 10.1016/0045-7825(82)90071-8
- T. J. R. Hughes, L. P. Franca, and M. Mallet, 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 thermodynamics, Comput. Methods Appl. Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553, DOI 10.1016/0045-7825(86)90127-1
- T. J. R. Hughes, L. P. Franca, and M. Mallet, 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 thermodynamics, Comput. Methods Appl. Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553, DOI 10.1016/0045-7825(86)90127-1
- Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0
- Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0 C. Johnson, U. Nävert, and J. Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), 285-312.
- Claes Johnson and Jukka Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp. 47 (1986), no. 175, 1–18. MR 842120, DOI 10.1090/S0025-5718-1986-0842120-4
- Claes Johnson and Anders Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), no. 180, 427–444. MR 906180, DOI 10.1090/S0025-5718-1987-0906180-5 —, On the convergence of streamline diffusion finite element methods for hyperbolic conservation laws, Numerical Methods for Compressible Flow—Finite Difference, Element and Volume Techniques (T. E. Tezduyar and T. J. H. Hughes, eds.), vol. 78, AMD. The American Society of Mechanical Engineers, 1986. —, Shock-capturing streamline diffusion finite element methods for nonlinear conservation laws, in Recent Developments in Computational Fluid Mechanics (T. E. Tezduyar and T. J. R. Hughes, eds.), vol. 95. AMD, The American Society of Mechanical Engineers, 1988.
- R. Löhner, K. Morgan, M. Vahdati, J. P. Boris, and D. L. Book, FEM-FCT: combining unstructured grids with high resolution, Comm. Appl. Numer. Methods 4 (1988), no. 6, 717–729. MR 969405, DOI 10.1002/cnm.1630040605
- Anders Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp. 53 (1989), no. 188, 527–545. MR 979941, DOI 10.1090/S0025-5718-1989-0979941-6
- Anders Szepessy, Measure-valued solutions of scalar conservation laws with boundary conditions, Arch. Rational Mech. Anal. 107 (1989), no. 2, 181–193. MR 996910, DOI 10.1007/BF00286499 —, Convergence of the streamline diffusion finite element method for conservation laws, Thesis, Dept. of Mathematics, Chalmers Univ. of Technology, S-412 96 Göteborg, 1989.
- Eitan Tadmor, Skew-selfadjoint form for systems of conservation laws, J. Math. Anal. Appl. 103 (1984), no. 2, 428–442. MR 762567, DOI 10.1016/0022-247X(84)90139-2
Additional Information
- © Copyright 1990 American Mathematical Society
- 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