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A linear backward Euler scheme for the saturation equation: regularity results and consistency. (English) Zbl 1273.65141

Summary: We consider a linearization of a numerical scheme for the saturation equation (or porous medium equation) \(\frac{\partial S}{\partial t} -\nabla \cdot f(S)\mathbf u-\nabla \cdot k(S)\nabla S = 0\), through first order expansions of the fractional function \(f\) and the inverse of the function \(K(s) = \int _0^s k(\tau )\, d\tau\), 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.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35Q35 PDEs in connection with fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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