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 f and the inverse of the function K ( s ) = ∫ 0 s k ( τ ) d τ , 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.