×
Szepessy, A.

Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions. (English) Zbl 0751.65061

RAIRO, Modélisation Math. Anal. Numér. 25, No. 6, 749-782 (1991).
The convergence of a higher-order accurate shock-capturing streamline diffusion finite element method is proved for general scalar conservation laws in several space dimensions with initial data and boundary conditions, using the uniqueness theorem for measure-valued solutions in the author’s paper [Arch. Ration. Mech. Anal. 107, No. 2, 181-193 (1989; Zbl 0702.35155)]. Numerical experiments are discussed.
Reviewer: V.A.Kostova (Russe)

Cited in 1 Review
Cited in 37 Documents

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws

Keywords:

convergence; shock-capturing streamline diffusion finite element method; conservation laws; Numerical experiments

Citations:

Zbl 0702.35155
PDFBibTeX XMLCite
\textit{A. Szepessy}, RAIRO, Modélisation Math. Anal. Numér. 25, No. 6, 749--782 (1991; Zbl 0751.65061)
Full Text: DOI EuDML

References:

[1] C. BARDOS, A. Y. LE ROUX, J. C. NEDELEC, First order quasilinear equations with boundary conditions, Comm. P.D.E. 4 (9), pp. 1017-1034, 1979. Zbl0418.35024 MR542510 · Zbl 0418.35024 · doi:10.1080/03605307908820117
[2] R. J. DIPERNA, Measure-valued solutions of conservation laws, Arch. Rat. Mech. Anal. 8 (1985). Zbl0616.35055 MR775191 · Zbl 0616.35055 · doi:10.1007/BF00752112
[3] R. J. DIPERNA, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82 (1983), 27-70. Zbl0519.35054 MR684413 · Zbl 0519.35054 · doi:10.1007/BF00251724
[4] F. DUBOIS et P. LE FLOCH, C. R. Acad., Sci. Paris 304, sériel (1987) 75-78. Zbl0634.35046 MR878830 · Zbl 0634.35046
[5] T. J. R. HUGHES and M. MALLET, A new finite element formulation for computational fluid dynamics : IV. a discontinuity - capturing operator for multidimensional advective - diffusive Systems, Comput. Methods Appl. Mech. Engrg. 58 (1986) 329-336. Zbl0587.76120 MR865672 · Zbl 0587.76120 · doi:10.1016/0045-7825(86)90153-2
[6] C. JOHNSON, U. NÄVERT and J. PITKÄRANTA, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984) 285-312. Zbl0526.76087 MR759811 · Zbl 0526.76087 · doi:10.1016/0045-7825(84)90158-0
[7] C. JOHNSON and J. SARANEN, Streamline diffusion methods for problems in fluid mechanics, Math. Comp. v. 47 (1986) pp.1-18. Zbl0609.76020 MR842120 · Zbl 0609.76020 · doi:10.2307/2008079
[8] C. JOHNSON and A. SZEPESSY, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp., vol.49, n^\circ 180, oct. 1987, pp. 427-444. Zbl0634.65075 MR906180 · Zbl 0634.65075 · doi:10.2307/2008320
[9] C. JOHNSON, A. SZEPESSY and P. HANSBOOn the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990) 82-107. Zbl0685.65086 MR995210 · Zbl 0685.65086 · doi:10.2307/2008684
[10] P. D. LAX, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E. A. Zarantonello, Academic Press (1971), 603-634. Zbl0268.35014 MR393870 · Zbl 0268.35014
[11] A. Y. LE ROUX, Étude du problème mixte pour une équation quasi linéaire du premier ordre, C. R. Acad. Sci. Paris, t. 285, Série A-351. Zbl0366.35019 MR442449 · Zbl 0366.35019
[12] A. Y. LE ROUX, Approximation de quelques problèmes hyperboliques non linéaires, Thèse d’État, Rennes, 1979.
[13] J. L. LIONS, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Paris, 1969. Zbl0189.40603 MR259693 · Zbl 0189.40603
[14] A. SZEPESSY, Convergence of a shock-capturing streamline diffusion finite element method for scalar conservation laws in two space dimensions, Math. Comp., Oct. 1989, 527-545. Zbl0679.65072 MR979941 · Zbl 0679.65072 · doi:10.2307/2008718
[15] A. SZEPESSY, An existence result for scalar conservation laws using measure valued solutions, Comm. PDE, 14 (10), 1989, 1329-1350. Zbl0704.35022 MR1022989 · Zbl 0704.35022 · doi:10.1080/03605308908820657
[16] A. SZEPESSY, Measure valued solutions of scalar conservation laws with boundary conditions, Arch. Rational Mech. Anal. 107, n^\circ 2, 1989, 181-193. Zbl0702.35155 MR996910 · Zbl 0702.35155 · doi:10.1007/BF00286499
[17] A. SZEPESSY, Convergence of the Streamline Diffusion Finite Element Method for Conservation Laws, Thesis (1989), Dept. of Math., Chalmers Univ., S-41296 Göteborg. · Zbl 0679.65072
[18] L. TARTAR, The Compensated Compactness Method Applied to Systems of Conservation Laws, J. M. Bail (ed.), Systems of Nonlinear Partial Differential Equations, 263-285. NATO ASI series C, Reidel Publishing Col. (1983). Zbl0536.35003 MR725524 · Zbl 0536.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.