×

zbMATH — the first resource for mathematics

Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions. (English) Zbl 0679.65072
A convergence result for a shock-capturing streamline diffusion finite element method applied to a time-dependent scalar nonlinear hyperbolic conservation law in two space dimensions is proved. The proof is based on a uniqueness result for measure-valued solutions by R. J. DiPerna [Arch. Ration. Mech. Anal. 88, 223-270 (1985; Zbl 0616.35055)]. It is also proved an almost optimal error estimate for a linearized conservation law having a smooth exact solution.
Reviewer: Ling Fuhua

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Citations:
Zbl 0616.35055
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223 – 270. · Zbl 0616.35055
[2] 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. , https://doi.org/10.1016/0045-7825(86)90152-0 T. J. R. Hughes and M. Mallet, Errata: ”A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems”, Comput. Methods Appl. Mech. Engrg. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes and Michel 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), no. 3, 329 – 336. , https://doi.org/10.1016/0045-7825(86)90153-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Marc Balestra, A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg. 59 (1986), no. 1, 85 – 99. , https://doi.org/10.1016/0045-7825(86)90025-3 T. J. R. Hughes, L. P. Franca, and M. Balestra, Errata: ”A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations”, Comput. Methods Appl. Mech. Engrg. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Michel Mallet, A new finite element formulation for computational fluid dynamics. VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 63 (1987), no. 1, 97 – 112. , https://doi.org/10.1016/0045-7825(87)90125-3 Thomas J. R. Hughes and Leopoldo P. Franca, A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg. 65 (1987), no. 1, 85 – 96. · Zbl 0635.76067
[3] 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. , https://doi.org/10.1016/0045-7825(86)90152-0 T. J. R. Hughes and M. Mallet, Errata: ”A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems”, Comput. Methods Appl. Mech. Engrg. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes and Michel 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), no. 3, 329 – 336. , https://doi.org/10.1016/0045-7825(86)90153-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Marc Balestra, A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg. 59 (1986), no. 1, 85 – 99. , https://doi.org/10.1016/0045-7825(86)90025-3 T. J. R. Hughes, L. P. Franca, and M. Balestra, Errata: ”A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations”, Comput. Methods Appl. Mech. Engrg. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Michel Mallet, A new finite element formulation for computational fluid dynamics. VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 63 (1987), no. 1, 97 – 112. , https://doi.org/10.1016/0045-7825(87)90125-3 Thomas J. R. Hughes and Leopoldo P. Franca, A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg. 65 (1987), no. 1, 85 – 96. · Zbl 0635.76067
[4] Claes Johnson, Uno Nävert, and Juhani Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1-3, 285 – 312. · Zbl 0526.76087
[5] Claes Johnson and Jukka Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp. 47 (1986), no. 175, 1 – 18. · Zbl 0609.76020
[6] 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. · Zbl 0634.65075
[7] Claes Johnson, Anders Szepessy, and Peter Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990), no. 189, 107 – 129. · Zbl 0685.65086
[8] S. N. Kružkov, ”First order quasilinear equations in several independent variables,” Math. USSR-Sb., v. 10, 1970, pp. 217-243.
[9] Anders Szepessy, Measure-valued solutions of scalar conservation laws with boundary conditions, Arch. Rational Mech. Anal. 107 (1989), no. 2, 181 – 193. · Zbl 0702.35155
[10] L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136 – 212. · Zbl 0437.35004
[11] Luc Tartar, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263 – 285. · 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.