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), 427444
MSC:
Primary 65M10; Secondary 65M60
MathSciNet review:
906180
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Abstract: We consider a spacetime finite element discretization of a timedependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). The finite element method is higherorder accurate and is a PetrovGalerkin method based on the socalled 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 MuratTartar, 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
(84k:35091), http://dx.doi.org/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 (84j:35114), http://dx.doi.org/10.1016/00219991(83)901183
 [3]
T.
J. R. Hughes and T.
E. Tezduyar, Finite element methods for firstorder hyperbolic
systems with particular emphasis on the compressible Euler equations,
Comput. Methods Appl. Mech. Engrg. 45 (1984),
no. 13, 217–284. MR 759810
(86a:65102), http://dx.doi.org/10.1016/00457825(84)901579
 [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. 761773.
 [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 NavierStokes
equations and the second law of thermodynamics, Comput. Methods Appl.
Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553
(87f:76010a), http://dx.doi.org/10.1016/00457825(86)901271
 [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
(87f:76010b), http://dx.doi.org/10.1016/00457825(86)901106
 [7]
Thomas
J. R. Hughes and Michel
Mallet, A new finite element formulation for computational fluid
dynamics. III. The generalized streamline operator for multidimensional
advectivediffusive systems, Comput. Methods Appl. Mech. Engrg.
58 (1986), no. 3, 305–328. MR 865671
(89j:76015a), http://dx.doi.org/10.1016/00457825(86)901520
 [8]
Thomas
J. R. Hughes and Michel
Mallet, A new finite element formulation for computational fluid
dynamics. IV. A discontinuitycapturing operator for multidimensional
advectivediffusive systems, Comput. Methods Appl. Mech. Engrg.
58 (1986), no. 3, 329–336. MR 865672
(89j:76015c), http://dx.doi.org/10.1016/00457825(86)901532
 [9]
Claes
Johnson, Uno
Nävert, and Juhani
Pitkäranta, Finite element methods for linear hyperbolic
problems, Comput. Methods Appl. Mech. Engrg. 45
(1984), no. 13, 285–312. MR 759811
(86a:65103), http://dx.doi.org/10.1016/00457825(84)901580
 [10]
Claes
Johnson and Jukka
Saranen, Streamline diffusion methods for the
incompressible Euler and NavierStokes equations, Math. Comp. 47 (1986), no. 175, 1–18. MR 842120
(88b:65133), http://dx.doi.org/10.1090/S00255718198608421204
 [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 FlowsFinite Difference, Element and Volume TechniquesAMD 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
(87m:35139)
 [14]
Eitan
Tadmor, Skewselfadjoint form for systems of conservation
laws, J. Math. Anal. Appl. 103 (1984), no. 2,
428–442. MR
762567 (86c:35100), http://dx.doi.org/10.1016/0022247X(84)901392
 [15]
L.
Tartar, Compensated compactness and applications to partial
differential equations, Nonlinear analysis and mechanics: HeriotWatt
Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston,
Mass., 1979, pp. 136–212. MR 584398
(81m:35014)
 [1]
 R. J. DiPerna, "Convergence of approximate solutions to conservation laws," Arch. Rational Mech. Anal., v. 82, 1983, pp. 2770. MR 684413 (84k:35091)
 [2]
 A. Harten, "On the symmetric form of systems of conservation laws with entropy," J. Comput. Phys., v. 49, 1983, pp. 151164. MR 694161 (84j:35114)
 [3]
 T. J. R. Hughes & T. E. Tezduyar, "Finite element methods for firstorder hyperbolic systems with particular emphasis on the compressible Euler equations," Comput. Methods Appl. Mech. Engrg., v. 45, 1984, pp. 217284. MR 759810 (86a:65102)
 [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. 761773.
 [5]
 T. J. R. Hughes, L. P. Franca & M. Mallet, "A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and NavierStokes equations and the second law of thermodynamics," Comput. Methods Appl. Mech. Engrg., v. 54, 1986, pp. 223234. MR 831553 (87f:76010a)
 [6]
 T. J. R. Hughes, M. Mallet & A. Mizukami, "A new finite element method for computational fluid dynamics: II. Beyond SUPG," Comput. Methods Appl. Mech. Engrg., v. 54, 1986, pp. 341355. MR 836189 (87f:76010b)
 [7]
 T. J. Hughes & M. Mallet, "A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advectivediffusive systems," Comput. Methods Appl. Mech. Engrg., v. 58, 1986, pp. 305328. MR 865671 (89j:76015a)
 [8]
 T. J. Hughes & M. Mallet, "A new finite element formulation for computational fluid dynamics: IV. A discontinuity capturing operator for multidimensional advectivediffusive systems," Comput. Methods Appl. Mech. Engrg., v. 58, 1986, pp. 329336. MR 865672 (89j:76015c)
 [9]
 C. Johnson, U. Nävert & J. Pitkäranta, "Finite element methods for linear hyperbolic problems," Comput. Methods Appl. Mech. Engrg., v. 45, 1985, pp. 285312. MR 759811 (86a:65103)
 [10]
 C. Johnson & J. Saranen, "Streamline diffusion methods for the incompressible Euler and NavierStokes equations," Math. Comp., v. 47, 1986, pp. 118. MR 842120 (88b:65133)
 [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 FlowsFinite Difference, Element and Volume TechniquesAMD vol. 78 (T. E. Tezduyar and T. J. R. Hughes, eds.).
 [13]
 J. Rauch, BV Estimates Fail for Most Quasilinear Hyperbolic Systems in Dimensions Greater Than One, Technical report, École Polytechnique, Palaiseau, 1986. MR 859822 (87m:35139)
 [14]
 E. Tadmor, "Skewselfadjoint forms for systems of conservation laws," J. Math. Anal. Appl., v. 103, 1984, pp. 428442. MR 762567 (86c:35100)
 [15]
 L. Tartar, "Compensated compactness and applications to partial differential equations," in Research Notes in Mathematics, Nonlinear Analysis and Mechanics: HeriotWatt Symposium, Vol. 4 (R. J. Knops, ed.), Pitman Press, London, 1979. MR 584398 (81m:35014)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198709061805
PII:
S 00255718(1987)09061805
Keywords:
Finite element method,
convergence,
nonlinear hyperbolic conservation law
Article copyright:
© Copyright 1987 American Mathematical Society
