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Mathematics of Computation
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
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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.


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Additional Information

Dongming Wei
Affiliation: Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148
Email: dwei@math.uno.edu

Lew Lefton
Affiliation: Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148
Email: llefton@math.uno.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01021-2
PII: S 0025-5718(99)01021-2
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