Numerical analysis of nonlinear elliptic-parabolic equations

Abstract

Abstract. This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996;Halpern, 1967;Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995;Kačur, 1999).Mathematics Subject Classification. 65M12, 35K65, 35K55, 65N22.

Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources

“…In fact, by a process of approximation, this assumption can be removed. Using the results in [21,26], we see that if f is sufficiently smooth, then so is the solution of the Eq. (3.50) with homogeneous Dirichlet boundary value condition.…”

Chronology of DIC technique based on the fundamental mathematical modeling and dehydration impact

“…Elliptic-parabolic problems have been applied in many applications, for example, as a model of flow through porous media (Bear 1975;Diaz and de Thelin 1994); pressure equation in an injection molding process (Maitre 2002); and also in electromagnetic field theory (MacCamy and Suri 1987). The results from Eqs.…”

Uniform-in-Time Convergence of Numerical Schemes for Richards’ and Stefan’s Models

“…This model includes both Richards' model (with ζ (s) = s), which describes the flow of water in an underground medium, and Stefan's model (with β (s) = s), which arises in the study of the heat diffusion in a melting medium. The numerical approximation of both Richards' and Stefan's models has been extensively studied in the literature (see the fundamental work on the Stefan problem [13], and [14] for a review of some numerical approximations, and see [12] for the Richards problem), but the convergence analysis of the considered schemes received a much reduced coverage and consists mostly in establishing space-time averaged (e.g. in L 2 (Ω × (0, T ))) results (in the case of finite volume schemes, see for example [6,9]).…”