A priori 𝐿^{𝜌} 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 \[ ∂ u ∂ t − ∑ i = 1 N ∂ ∂ x i ( | u | ρ − 2 ∂ u ∂ x i ) = f ( x , t ) \frac {\partial u}{\partial t}-\sum ^N_{i=1}\frac \partial {\partial x_i}(|u|^{\rho -2}\frac {\partial u}{ \partial x_i})=f(x,t) \] on bounded convex domains are considered. The range of the parameter ρ \rho includes the fast diffusion case 1 > ρ > 2 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 ∞ ( 0 , T ; L ρ ( Ω ) ) L^\infty (0,T;L^\rho (\Omega )) norm with an error controlled by O ( Δ t 1 4 ) O(\Delta t^{\frac 14}) for 1 > ρ > 2 1>\rho >2 and O ( Δ t 1 2 ρ ) O(\Delta t^{\frac 1{2\rho }}) for 2 ≤ ρ > ∞ 2\le \rho >\infty . For the fully discrete problem, a global convergence rate of O ( Δ t 1 4 ) O(\Delta t^{\frac 14}) in L 2 ( 0 , T ; L ρ ( Ω ) ) L^2(0,T;L^\rho (\Omega )) norm is shown for the range 2 N N + 1 > ρ > 2 \frac {2N}{N+1}>\rho >2 . For 2 ≤ ρ > ∞ 2\le \rho >\infty , a rate of O ( Δ t 1 2 ρ ) O(\Delta t^{\frac 1{2\rho }}) is shown in L ρ ( 0 , T ; L ρ ( Ω ) ) L^\rho (0,T;L^\rho (\Omega )) norm.