A linear backward Euler scheme for the saturation equation: Regularity results and consistency

We consider a linearization of a numerical scheme for the saturation equation (or porous medium equation) ∂S∂t−∇⋅f(S)u−∇⋅k(S)∇S=0, through first order expansions of the fractional function ff and the inverse of the function K(s)=∫0sk(τ)dτ, after a regularization of the porous medium equation. We establish a regularity result for the Continuous Galerkin Method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme. We then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work.