Convergence of a shockcapturing streamline diffusion finite element method for a scalar conservation law in two space dimensions
Author:
Anders Szepessy
Journal:
Math. Comp. 53 (1989), 527545
MSC:
Primary 65M60
MathSciNet review:
979941
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Abstract: We prove a convergence result for a shockcapturing streamline diffusion finite element method applied to a timedependent scalar nonlinear hyperbolic conservation law in two space dimensions. The proof is based on a uniqueness result for measurevalued solutions by DiPerna. We also prove an almost optimal error estimate for a linearized conservation law having a smooth exact solution.
 [1]
Ronald
J. DiPerna, Measurevalued solutions to conservation laws,
Arch. Rational Mech. Anal. 88 (1985), no. 3,
223–270. MR
775191 (86g:35121), http://dx.doi.org/10.1007/BF00752112
 [2]
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
 [3]
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 timedependent multidimensional advectivediffusive systems,
Comput. Methods Appl. Mech. Engrg. 63 (1987), no. 1,
97–112. MR
896773 (89j:76015f), http://dx.doi.org/10.1016/00457825(87)901253
 [4]
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
 [5]
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
 [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. MR 906180
(88h:65164), http://dx.doi.org/10.1090/S00255718198709061805
 [7]
Claes
Johnson, Anders
Szepessy, and Peter
Hansbo, On the convergence of shockcapturing
streamline diffusion finite element methods for hyperbolic conservation
laws, Math. Comp. 54
(1990), no. 189, 107–129. MR 995210
(90j:65118), http://dx.doi.org/10.1090/S00255718199009952100
 [8]
S. N. Kružkov, "First order quasilinear equations in several independent variables," Math. USSRSb., v. 10, 1970, pp. 217243.
 [9]
Anders
Szepessy, Measurevalued solutions of scalar conservation laws with
boundary conditions, Arch. Rational Mech. Anal. 107
(1989), no. 2, 181–193. MR 996910
(90f:35129), http://dx.doi.org/10.1007/BF00286499
 [10]
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.London, 1979, pp. 136–212. MR 584398
(81m:35014)
 [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. MR 725524
(85e:35079)
 [1]
 R. J. DiPerna, "Measurevalued solutions of conservation laws," Arch. Rational Mech. Anal., v. 88, 1985, pp. 223270. MR 775191 (86g:35121)
 [2]
 T. J. R. Hughes & M. Mallet, "A new finite element formulation for computational fluid dynamics: IV. A discontinuitycapturing operator for multidimensional advectivediffusive systems," Comput. Methods Appl. Mech. Engrg., v. 58, 1986, pp. 329336. MR 865672 (89j:76015c)
 [3]
 T. J. R. Hughes, L. P. Franca & M. Mallet, "VI. Convergence analysis of the generalized SUPG formulation for linear timedependent multidimensional advectivediffusive systems," Comput. Methods Appl. Mech. Engrg., v. 63, 1987, pp. 97112. MR 896773 (89j:76015f)
 [4]
 C. Johnson, U. Nävert & J. Pitkäranta, "Finite element methods for linear hyperbolic problems," Comput. Methods Appl. Mech. Engrg., v. 45, 1984, pp. 285312. MR 759811 (86a:65103)
 [5]
 C. Johnson & J. Saranen, "Streamline diffusion methods for problems in fluid mechanics," Math. Comp., v. 47, 1986, pp. 118. MR 842120 (88b:65133)
 [6]
 C. Johnson & A. Szepessy, "On the convergence of a finite element method for a nonlinear hyperbolic conservation law," Math. Comp., v. 49, 1987, pp. 427444. MR 906180 (88h:65164)
 [7]
 C. Johndon, A. Szepessy & P. Hansbo, On the Convergence of ShockCapturing Streamline Diffusion Finite Element Methods for Hyperbolic Conservation Laws, Preprint No 1987:21, Dept. of Math., Chalmers Univ. of Technology, S412 96 Göteborg. (To appear in Math. Comp.) MR 995210 (90j:65118)
 [8]
 S. N. Kružkov, "First order quasilinear equations in several independent variables," Math. USSRSb., v. 10, 1970, pp. 217243.
 [9]
 A. Szepessy, "Measurevalued solutions of scalar conservation laws with boundary conditions," Arch. Rational Mech. Anal. (To appear.) MR 996910 (90f:35129)
 [10]
 L. Tartar, Compensated Compactness and Applications to Partial Differential Equations, in Research Notes in Mathematics, Nonlinear Analysis and Mechanics: HeriotWatts Symposium, vol. 4 (R. J. Knops, ed.), Pitman Press, London, 1979. MR 584398 (81m:35014)
 [11]
 L. Tartar, The Compensated Compactness Method Applied to Systems of Conservation Laws, Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), NATO ASI series, Reidel, Dordrecht, 1983, pp. 263285. MR 725524 (85e:35079)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909799416
PII:
S 00255718(1989)09799416
Article copyright:
© Copyright 1989
American Mathematical Society
