Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A priori $L^\rho$ error estimates
for Galerkin approximations
to porous medium and fast diffusion equations

Authors: Dongming Wei and Lew Lefton
Journal: Math. Comp. 68 (1999), 971-989
MSC (1991): Primary 65M60, 35K60, 35K65
Published electronically: February 11, 1999
MathSciNet review: 1609654
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation

\begin{displaymath}\frac{\partial u}{\partial t}-\sum^N_{i=1}\frac \partial{\partial x_i}(|u|^{\rho-2}\frac{\partial u}{ \partial x_i})=f(x,t)\end{displaymath}

on bounded convex domains are considered. The range of the parameter $\rho$ includes the fast diffusion case $1<\rho<2$. Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in $L^\infty(0,T;L^\rho(\Omega))$ norm with an error controlled by $O(\Delta t^{\frac 14})$ for $1<\rho<2$ and $O(\Delta t^{\frac 1{2\rho}})$ for $2\le \rho<\infty$. For the fully discrete problem, a global convergence rate of $O(\Delta t^{\frac 14})$ in $L^2(0,T;L^\rho(\Omega))$ norm is shown for the range $\frac {2N}{N+1}<\rho<2$. For $2\le \rho<\infty$, a rate of $O(\Delta t^{\frac 1{2\rho}})$ is shown in $L^\rho(0,T;L^\rho(\Omega))$ norm.

References [Enhancements On Off] (What's this?)

  • 1. J. G. Berryman and C. J. Holland, Nonlinear diffusion problem arising in plasma physics, Phys. Rev. Lett. 40 (1978), 1720-1722. MR 58:14366
  • 2. H. Brézis and M. Crandall, Uniqueness of solutions of the initial value problem for $u_t-\Delta \phi(u)=0$, J. Math. Pures Appl. 58 (1979), 153-163. MR 80e:35029
  • 3. H. Brézis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), 73-97. MR 84g:35093
  • 4. F. E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, in Proceedings of Symposia in Applied Mathematics 17, Amer. Math. Soc., Providence, RI, 1964. MR 33:6092
  • 5. P. G. Ciarlet and J. L. Lions (eds.), Handbook of Numerical Analysis, Volume II: Finite Element Methods (Part 1), North-Holland, 1991. MR 92f:65001
  • 6. E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Berlin, 1994. MR 94h:35130
  • 7. T. DuPont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), 441-463. MR 81h:65014
  • 8. A. Eden, B. Michaux, and J. M. Rakotoson, Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis, Indiana U. Math. J. 39 (1990), 737-783. MR 91h:35150
  • 9. C. M. Elliott, Error analysis of the enthalpy method for the Stefan problem, IMA J. of Num. Anal. 7 (1987), 61-71. MR 90a:65222
  • 10. I. Farago, Finite element method for solving nonlinear parabolic systems, Computers & Math. with Appl. 21 (1991), 49-59.
  • 11. S. M. F. Garcia, Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: The discrete-time case, Num. Meth. for Part. Diff. Eq. 10 (1994), 149-170. MR 95a:65164
  • 12. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, New York, 1983. MR 86c:35035
  • 13. J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Royal Soc. Lond. A 343 (1993), 337-375.
  • 14. J. W. Jerome and M. E. Rose, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp. 39 (1982), 377-414. MR 84h:65097
  • 15. A. S. Kalashnikov, Some problems of the qualitative theory of the non-linear degenerate second order parabolic equations, Russian Math. Surveys 42 (1987), 169-222.
  • 16. M.-N. Le Roux, Semidiscretization in time of a fast diffusion equation, J. Math. Anal. Appl. 137 (1989), 354-370. MR 90k:65166
  • 17. P. Lesaint and J. Pousin, Error estimates for a nonlinear degenerate parabolic problem, Math. Comp. 59 (1992), 339-358. MR 93a:35094
  • 18. J. L. Lions, Quelqes méthodes de résolution des problemès aux limites non linéaires, Dunod, Paris, 1969. MR 41:4326
  • 19. R. H. Nochetto, Error estimates for multidimensional singular parabolic problems, Japan J. Appl. Math. 4 (1987), 111-138. MR 89c:65107
  • 20. R. H. Nochetto and C. Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal. 25 (1988), 784-814. MR 89m:65102
  • 21. R. H. Nochetto, personal communication.
  • 22. P. A. Raviart, Sur la résolution de certaines équations paraboliques non linéaires dégénérées, J. Func. Anal. 5 (1970), 299-328. MR 41:2235
  • 23. P. A. Raviart, Sur la résolution et l'approximation de certaines équations paraboliques non linéaires dégénerées, Arch. Rat. Mech. Anal. 25 (1967), 64-80. MR 35:6384
  • 24. M. E. Rose, Numerical methods for flows through porous media. I, Math. Comp. 40 (1983), 435-467. MR 85a:65146
  • 25. J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems, SIAM J. on Numer. Anal. 33 (1996), 68-87. MR 97c:65151
  • 26. J. Rulla and N. Walkington, Optimal rates of convergence for degenerate parabolic problems in two dimensions, SIAM J. on Numer. Anal. 33 (1996), 56-67. MR 97c:65150
  • 27. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), 483-493. MR 90j:65021
  • 28. M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. and Appl. 132 (1988), 187-212. MR 89k:35118
  • 29. C. Verdi, Optimal error estimates for an approximation of degenerate parabolic problems, Numer. Funct. Anal. Optim. 9 (1987), 657-670. MR 88m:65165
  • 30. A. \v{Z}ení\v{s}ek, Nonlinear elliptic and evolution problems and their finite element approximations, Academic Press, New York, 1990. MR 92c:65003

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65M60, 35K60, 35K65

Retrieve articles in all journals with MSC (1991): 65M60, 35K60, 35K65

Additional Information

Dongming Wei
Affiliation: Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148

Lew Lefton
Affiliation: Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148

Keywords: Porous medium equation, fast diffusion equation, Cauchy-Dirichlet problem, finite elements, $L^\rho$ error estimates, Galerkin approximations
Received by editor(s): April 17, 1996
Received by editor(s) in revised form: October 22, 1997
Published electronically: February 11, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society