SUMMARYIn this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, allows us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is established in terms of the mesh size and the properties of the material. In the one-dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization technique.

This paper deals with the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain Ω. In a region Ωp a diffusion-advection-reaction type equation is set, while in the complementary Ωh ≡ Ω\Ωp, only advection-reaction terms are taken into account. To begin we provide a definition of a weak solution through an entropy inequality on the whole domain. Since the interface ∂Ωp ∩ ∂Ωh contains outward characteristics for the first-order operator in Ωh, the uniqueness proof starts by considering first the hyperbolic zone and then the parabolic one. The existence property uses the vanishing viscosity method and to pass to the limit on the hyperbolic zone, we refer to the notion of process solution.

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We consider a kind of singularly perturbed problem with a small positive parameter affecting the second order derivative only in a part of the domain. We analyse the existence and uniqueness of the solution and the asymptotic behaviour as the small parameter goes to zero. ᮊ 1997 Academic Press