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

Galerkin approximations to solutions of a Caucliy-Dirichlet problem governed by the generalized porous medium equation (formula presented) on bounded convex domains are considered. The range of the parameter ρ includes the fast diffusion case 1 < ρ < 2. Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in L^∞(0, T; L^ρ(Ω)) norm with an error controlled by O(Δt1/1) for 1 < p < 2 and O(Δt1/2ρ) for 2 ≤ p < ∞. For the fully discrete problem, a global convergence rate of O(Δt1/4) in L²(0, T; L^ρ(Ω)) norm is shown for the range 2N/N+1 < ρ < 2. For 2 ≤ ρ < ∞, a rate of O(At1/2ρ) is shown in L^ρ(0, T; L^ρ(Ω)) norm.