Convergence Estimates for a Linear Backward Euler Scheme for the Saturation Equation

ABSTRACT

In a previous work, stability and consistency results were established for a linearized Euler scheme for the saturation equation. In this paper we continue the mathematical analysis of the scheme, in preparation for its numerical treatment in a future work. We use the regularity results, obtained previously, to establish error estimates in L² (Ω) for the linear scheme. This work is done with the degenerate nature of the saturation equation in mind, but it is also valid for the non degenerate case like the concentration equation. We show that, if the regularization parameter β and the spatial discretization parameter h are carefully chosen in terms of the time stepping parameter Δt , the convergence is at least of order 0((Δt )α ) for some determined α > 0 . Examples of choices of β and h are given. We also establish a new (at our knowledge) regularity result for the continuous Galerkin formulation of the Saturation Equation and a new regularity result for the linear scheme.

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