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

Abstract:

Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation \[ \frac {\partial u}{\partial t}-\sum ^N_{i=1}\frac \partial {\partial x_i}(|u|^{\rho -2}\frac {\partial u}{ \partial x_i})=f(x,t)\] 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.