Stability and convergence of a class of finite element schemes for hyperbolic systems of conservation laws

SIAM Journal on Numerical Analysis

Volumn 42, Issue 4, 2004, Pages 1357-1393

  Arvanitis, Christos ^a,b   Makridakis, Charalambos ^a,b   Tzavaras, Athanasios E ^b,c  

^a UNIVERSITY OF CRETE   (Greece)

^b INSTITUTE OF COMPUTER SCIENCE   (Greece)

^c UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Adaptive schemes; Finite element schemes; Hyperbolic conservation laws; Stability and convergence

Indexed keywords

ADAPTIVE SCHEMES; FINITE ELEMENT SCHEMES; HYPERBOLIC CONSERVATION LAWS; STABILITY AND CONVERGENCE;

CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; MATHEMATICAL MODELS; POLYNOMIALS;

FINITE ELEMENT METHOD;

EID: 25444433067     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902420436     Document Type: Article

References (43)

1. D. AREGBA-DRIOLLET AND R. NATALINI, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal., 37 (2000), pp. 1973-2004.
2. CH. ARVANITIS, Finite Elements for Hyperbolic Systems of Conservation Laws: New Methods and Computational Techniques, Ph.D. thesis, University of Crete, Crete, Greece, 2002.
3. CH. ARVANITIS, Mesh Refinement Strategies and Finite Element Schemes for Hyperbolic Systems of Conservation Laws, preprint, University of Crete, Crete, Greece, 2004.
4. CH. ARVANITIS, TH. KATSAOUNIS, AND CH. MAKRIDAKIS, Adaptive finite element relaxation schemes for hyperbolic conservation laws, Math. Model. Numer. Anal., 35 (2001), pp. 17-33.
5. I. BABUSKA AND J. E. OSBORN, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math., 34 (1980), pp. 41-62.
6. I. BABUŠKA, J. E. OSBORN, AND J. PITKÄRANTA, Analysis of mixed methods using mesh dependent norms, Math. Comp., 35 (1980), pp. 1039-1062.
7. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
8. G.-Q. CHEN, C. D. LEVERMORE, AND T.-P. LIU, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), pp. 789-830.
9. B. COCKBURN AND P.-A. GREMAUD, Error estimates for finite element methods for scalar conservation laws, SIAM J. Numer. Anal., 33 (1996), pp. 522-554.
10. B. COCKBURN, S. HOU, AND C.-W. SHU, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp., 54 (1990), pp. 545-581.
11. B. COCKBURN, C. JOHNSON, C.-W. SHU, AND E. TADMOR, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, A. Quarteroni, ed., Lecture Notes in Math. 1697, Springer-Verlag, New York, 1998.
12. B. COCKBURN AND C.-W. SHU, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp., 52 (1989), pp. 411-435.
13. 2 projection into finite element function spaces, Math. Comp., 48 (1987), pp. 521-532.
14. C.M. DAFERMOS, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss. 325, Springer-Verlag, Berlin, 2000.
15. R. DIPERNA, Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal., 60 (1983), pp. 75-100.
16. ∞, SIAM J. Numer. Anal., 32 (1995), pp. 706-740.
17. E. GODLEWSKI AND P.-A. RAVIART, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math. Ser. 118, Springer-Verlag, Berlin, 1996.
18. L. GOSSE AND A. TZAVARAS, Convergence of relaxation schemes to the equations of elastodynamics, Math. Comp., 70 (2001), pp. 555-577.
19. J. JAFFRÉ, C. JOHNSON, AND A. SZEPESSY, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Models Methods Appl. Sci., 5 (1995), pp. 367-386.
20. S. JIN AND Z. XIN, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), pp. 235-277.
21. C. JOHNSON AND A. SZEPESSY, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp., 49 (1987), pp. 427-444.
22. C. JOHNSON AND A. SZEPESSY, Adaptive finite element methods for conservation laws. Part I: The general approach, Comm. Pure Appl. Math., 48 (1995), pp. 199-234.
23. C. JOHNSON, A. SZEPESSY, AND P. HANSBO, On the convergence of shock-capturing stream-line diffusion finite element methods for hyperbolic conservation laws, Math. Comp., 54 (1990), pp. 107-129.
24. T. KATSAOUNIS AND CH. MAKRIDAKIS, Finite volume relaxation schemes for multidimensional conservation laws, Math. Comp., 70 (2001), pp. 533-553.
25. M.A. KATSOULAKIS, G.T. KOSSIORIS, AND CH. MAKRIDAKIS, Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Comm. Partial Differential Equations, 24 (1999), pp. 395-424.
26. D. KRÖNER, Numerical Schemes for Conservation Laws, John Wiley, New York, 1997.
27. A. KURGANOV AND E. TADMOR, New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), pp. 241-282.
28. PH.G. LEFLOCH, Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves, Lectures in Mathematics ETH Zurich, Birkhäuser Verlag, Basel, Switzerland, 2002.
29. PH.G. LEFLOCH AND C. ROHDE, High-order schemes, entropy inequalities, and nonclassical shocks, SIAM J. Numer. Anal., 37 (2000), pp. 2023-2060.
30. R.J. LEVEQUE, Numerical Methods for Conservation Laws, Lectures Math. ETH Zurich, Birkhäuser, Zurich, 1992.
31. P. LIN, Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics, Trans. Amer. Math. Soc., 329 (1992), pp. 377-413.
32. -1,q est compacte pour tout q < 2, J. Math. Pures Appl., 60 (1981), pp. 309-322.
33. H. NESSYAHU AND E. TADMOR, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408-463.
34. B. PERTHAME AND A.E. TZAVARAS, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch. Ration. Mech. Anal., 155 (2000), pp. 1-48.
35. M.E. SCHONBEK, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), pp. 959-1000.
36. D. SERRE, Relaxation semi linéaire et cinétique des systèmes de lois de conservation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), pp. 169-192.
37. D. SERRE AND J. SHEARER, Convergence with Physical Viscosity for Nonlinear Elasticity, manuscript, 1993.
38. p for the quasi-linear wave equation, Comm. Partial Differential Equations, 19 (1994), pp. 1829-1877.
39. C.-W. SHU, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 1073-1084.
40. C.-W. SHU AND S. OSHER, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439-471.
41. A. SZEPESSY, Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp., 53 (1989), pp. 527-545.
42. L. TARTAR, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, R. J. Knops, ed., Res. Notes in Math. 39, Pitman, Boston, 1979, pp. 136-212.
43. A. TZAVARAS, Materials with internal variables and relaxation to conservation laws, Arch. Ration. Mech. Anal., 146 (1999), pp. 129-155.