"Convergence of Finite Volume Approximations for a Nonlinear Elliptic-Parabolic Problem: a ""Continuous"" Approach."

Boris Andreianov (1) , Michaël Gutnic (2) , Petra Wittbold (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 IRMA - Institut de Recherche Mathématique Avancée

"We study the approximation by finite volume methods of the modelparabolic-elliptic problem $b(v)_t=div (|Dv|^{p-2} Dv)$ on$(0,T)imesOmegasubset RimesR^d$ with an initial conditionand the homogeneous Dirichlet boundary condition. Because of thenonlinearity in the elliptic term, a careful choice of the gradient approximation is needed.We prove the convergence of discrete solutions to a solution ofthe continuous problem as the discretization step $h$ tends to $0$,under the main hypotheses that the approximation of the operator$div(|Dv|^{p-2} Dv)$ provided by the finite volume scheme is still monotone and coercive, and that the gradient approximationis exact on the affine functions of $xinOmega$. An example of such a scheme is given for a class of two-dimensional meshes dual to triangular meshes, in particular for structured rectangular andhexagonal meshes.The proof uses the rewriting of the discrete problem under a``continuous'' form. This permits us to directly apply theAlt-Luckhaus variational techniques known in the continuous case. "

Mots clés: convergence of approximate solutions; "doubly nonlinear elliptic-parabolic equations; finite volume methods; doubly nonlinear elliptic-parabolic equations; continuous approach; continuous approach"

Domaines: Analyse numérique [math.NA]

Boris Andreianov, Michaël Gutnic, Petra Wittbold. "Convergence of Finite Volume Approximations for a Nonlinear Elliptic-Parabolic Problem: a ""Continuous"" Approach.". SIAM Journal on Numerical Analysis, 2003, 42 (1), pp.228-251. ⟨hal-00129680⟩