On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws

Authors:
Claes Johnson, Anders Szepessy and Peter Hansbo

Journal:
Math. Comp. **54** (1990), 107-129

MSC:
Primary 65M60; Secondary 35L65, 76L05

DOI:
https://doi.org/10.1090/S0025-5718-1990-0995210-0

MathSciNet review:
995210

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock-capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the mesh size. With this term present, we prove a maximum norm bound for finite element solutions of Burgers' equation and thus complete an earlier convergence proof for this equation. We further prove, using entropy variables, that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality associated with the entropy variables. Results of some numerical experiments for the time-dependent compressible Euler equations in two dimensions are also reported.

**[1]**K. Eriksson and C. Johnson,*An adaptive finite element method for linear elliptic problems*, Math. Comp.**50**(1988), 361-383. MR**929542 (89c:65119)****[2]**P. Hansbo,*Finite element procedures for conduction and convection problems*, Publication 86:7, Dept. of Structural Mechanics, Chalmers Univ. of Technology, S-412 96 Göteborg, 1986.**[3]**-,*Streamline diffusion methods and adaptive procedures in finite element methods*, Thesis, Dept. of Structural Mechanics, Chalmers Univ. of Technology, 1989.**[4]**A. Harten,*On the symmetric form of systems of conservation laws with entropy*, Comput. Phys.**49**(1983), 151-164. MR**694161 (84j:35114)****[5]**T. J. R. Hughes and A. Brook,*Streamline upwind-Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations*, Comput. Methods Appl. Mech. Engrg.**32**(1982), 199-259. MR**679322 (83k:76005)****[6]**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), 223-234. MR**831553 (87f:76010a)****[7]**T. J. R. Hughes, M. Mallet, and A. Mizukami,*A new finite element formulation for computational fluid dynamics*: II.*Beyond SUPG*, Comput. Methods Appl. Mech. Engrg.**54**(1986), 341-355. MR**836189 (87f:76010b)****[8]**T. J. R. Hughes and M. Mallet,*A new finite element formulation for computational fluid dynamics*: III.*The general streamline operator for multidimensional advective-diffusive systems*, Comput. Methods Appl. Mech. Engrg.**58**(1986), 305-328. MR**865671 (89j:76015a)****[9]**-,*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), 329-336. MR**865672 (89j:76015c)****[10]**C. Johnson, U. Nävert, and J. Pitkäranta,*Finite element methods for linear hyperbolic problems*, Comput. Methods Appl. Mech. Engrg.**45**(1984), 285-312.**[11]**C. Johnson and J. Saranen,*Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations*, Math. Comp.**47**(1986), 1-18. MR**842120 (88b:65133)****[12]**C. Johnson and A. Szepessy,*On the convergence of a finite element method for a nonlinear hyperbolic conservation law*, Math. Comp.**49**(1987), 427-444. MR**906180 (88h:65164)****[13]**-,*On the convergence of streamline diffusion finite element methods for hyperbolic conservation laws*, Numerical Methods for Compressible Flow--Finite Difference, Element and Volume Techniques (T. E. Tezduyar and T. J. H. Hughes, eds.), vol. 78, AMD. The American Society of Mechanical Engineers, 1986.**[14]**-,*Shock-capturing streamline diffusion finite element methods for nonlinear conservation laws*, in Recent Developments in Computational Fluid Mechanics (T. E. Tezduyar and T. J. R. Hughes, eds.), vol. 95. AMD, The American Society of Mechanical Engineers, 1988.**[15]**R. Löhner, K. Morgan, and M. Vahdati,*FEM-FCT: Combining unstructured grids with high resolution*, Comm. Appl. Numer. Methods**4**(1988), 717-729. MR**969405****[16]**A. Szepessy,*Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions*, Math. Comp.**53**(1989), 527-545. MR**979941 (90h:65156)****[17]**-,*Measure valued solutions to scalar conservation laws with boundary conditions*, Arch. Rational Mech. Anal. (to appear). MR**996910 (90f:35129)****[18]**-,*Convergence of the streamline diffusion finite element method for conservation laws*, Thesis, Dept. of Mathematics, Chalmers Univ. of Technology, S-412 96 Göteborg, 1989.**[19]**E. Tadmor,*Skew-selfadjoint forms for systems of conservation laws*, J. Math. Anal. Appl.**103**(1984), 428-442. MR**762567 (86c:35100)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M60,
35L65,
76L05

Retrieve articles in all journals with MSC: 65M60, 35L65, 76L05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-0995210-0

Keywords:
Finite element method,
conservation laws,
convergence streamline diffusion,
shock-capturing,
entropy variables

Article copyright:
© Copyright 1990
American Mathematical Society