Adaptive finite element methods for conservation laws based on a posteriori error estimates

We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way. ©1995 John Wiley & Sons, Inc.

[1]  Tamir Tassa,et al.  Convergence rate of approximate solutions to conservation laws with initial rarefactions , 1994 .

[2]  Peter Hansbo,et al.  Space-time oriented streamline diffusion methods for nonlinear conservation laws in one dimension , 1994 .

[3]  Kenneth Eriksson,et al.  Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems , 1993 .

[4]  Claes Johnson,et al.  A new approach to algorithms for convection problems which are based on exact transport + projection , 1992 .

[5]  Peter Hansbo,et al.  The characteristic streamline diffusion method for the time-dependent incompressible Navier-Stokes equations , 1992 .

[6]  Z. Xin,et al.  Viscous limits for piecewise smooth solutions to systems of conservation laws , 1992 .

[7]  A. Szepessy.  On the stability of finite element methods for shock waves , 1992 .

[8]  Peter Hansbo,et al.  The characteristic streamline diffusion method for convection-diffusion problems , 1992 .

[9]  E. Tadmor.  Local error estimates for discontinuous solutions of nonlinear hyperbolic equations , 1991 .

[10]  Peter Hansbo,et al.  Adaptive streamline diffusion methods for compressible flow using conservation variables , 1991 .

[11]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

[12]  Zhouping Xin,et al.  Nonlinear stability of viscous shock waves , 1993 .

[13]  Peter Hansbo,et al.  A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equation , 1990 .

[14]  Claes Johnson,et al.  Adaptive finite element methods for diffusion and convection problems , 1990 .

[15]  Peter Hansbo,et al.  On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws , 1990 .

[16]  Anders Szepessy,et al.  Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions , 1989 .

[17]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[18]  Eduard Harabetian,et al.  Rarefactions and large time behavior for parabolic equations and monotone schemes , 1988 .

[19]  Asymptotic Stability of Rarefaction Waves for 2 * 2 Viscous Hyperbolic Conservation Laws , 1988 .

[20]  Claes Johnson,et al.  On the convergence of a finite element method for a nonlinear hyperbolic conservation law , 1987 .

[21]  Jonathan Goodman,et al.  Nonlinear asymptotic stability of viscous shock profiles for conservation laws , 1986 .

[22]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems , 1986 .

[23]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems , 1986 .

[24]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[25]  B. Lucier.  A moving mesh numerical method for hyperbolic conservation laws , 1986 .

[26]  Tai-Ping Liu,et al.  Nonlinear Stability of Shock Waves for Viscous Conservation Laws , 1985 .

[27]  R. Pego.  Remarks on the stability of shock profiles for conservation laws with dissipation , 1985 .

[28]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[29]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .