Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions

Szepessy, Anders

doi:10.1090/S0025-5718-1989-0979941-6

Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions
HTML articles powered by AMS MathViewer

by Anders Szepessy PDF

Math. Comp. 53 (1989), 527-545 Request permission

Abstract:

We prove 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. The proof is based on a uniqueness result for measure-valued solutions by DiPerna. We also prove an almost optimal error estimate for a linearized conservation law having a smooth exact solution.

References

Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI 10.1007/BF00752112
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
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. MR 759811, DOI 10.1016/0045-7825(84)90158-0
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
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. MR 995210, DOI 10.1090/S0025-5718-1990-0995210-0

Math. USSR-Sb.

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
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. MR 584398
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. MR 725524