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

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

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.