On the convergence of a finite element method for a nonlinear hyperbolic conservation law

Authors:
Claes Johnson and Anders Szepessy

Journal:
Math. Comp. **49** (1987), 427-444

MSC:
Primary 65M10; Secondary 65M60

MathSciNet review:
906180

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function *u*, then *u* is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers' equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments.

**[1]**R. J. DiPerna,*Convergence of approximate solutions to conservation laws*, Arch. Rational Mech. Anal.**82**(1983), no. 1, 27–70. MR**684413**, 10.1007/BF00251724**[2]**Amiram Harten,*On the symmetric form of systems of conservation laws with entropy*, J. Comput. Phys.**49**(1983), no. 1, 151–164. MR**694161**, 10.1016/0021-9991(83)90118-3**[3]**T. J. R. Hughes and T. E. Tezduyar,*Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations*, Comput. Methods Appl. Mech. Engrg.**45**(1984), no. 1-3, 217–284. MR**759810**, 10.1016/0045-7825(84)90157-9**[4]**T. J. Hughes, M. Mallet & L. P. Franca, "Entropy stable finite element methods for compressible fluids: Application to high Mach number flows with shocks," in*Finite Elements for Nonlinear Problems*(P. Bergen, K. J. Bathe and W. Wunderlich, eds.), Springer, Berlin, 1986, pp. 761-773.**[5]**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**, 10.1016/0045-7825(86)90127-1**[6]**Thomas J. R. Hughes, Michel Mallet, and Akira Mizukami,*A new finite element formulation for computational fluid dynamics. II. Beyond SUPG*, Comput. Methods Appl. Mech. Engrg.**54**(1986), no. 3, 341–355. MR**836189**, 10.1016/0045-7825(86)90110-6**[7]**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**, 10.1016/0045-7825(86)90152-0**[8]**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. MR**865672**, 10.1016/0045-7825(86)90153-2**[9]**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**, 10.1016/0045-7825(84)90158-0**[10]**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**, 10.1090/S0025-5718-1986-0842120-4**[11]**C. Johnson, "Streamline diffusion methods for problems in fluid mechanics," in*Finite Elements in Fluids*, vol. 6 (Gallagher et al., eds.), Wiley, New York, 1985.**[12]**C. Johnson & A. Szepessy,*On the Convergence of Streamline Diffusion Finite Element Methods for Hyperbolic Conservation Laws*, Numerical Methods for Compressible Flows--Finite Difference, Element and Volume Techniques--AMD vol. 78 (T. E. Tezduyar and T. J. R. Hughes, eds.).**[13]**Jeffrey Rauch,*BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one*, Comm. Math. Phys.**106**(1986), no. 3, 481–484. MR**859822****[14]**Eitan Tadmor,*Skew-selfadjoint form for systems of conservation laws*, J. Math. Anal. Appl.**103**(1984), no. 2, 428–442. MR**762567**, 10.1016/0022-247X(84)90139-2**[15]**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**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M10,
65M60

Retrieve articles in all journals with MSC: 65M10, 65M60

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1987-0906180-5

Keywords:
Finite element method,
convergence,
nonlinear hyperbolic conservation law

Article copyright:
© Copyright 1987
American Mathematical Society