Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities

Franck Boyer (1) , Florence Hubert (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

In this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the Discrete Duality Finite Volume framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems.

Mots clés: Nonlinear elliptic problems; Finite Volume schemes; Discontinuous coefficients; Nonlinear elliptic problems.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Franck Boyer, Florence Hubert. Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM Journal on Numerical Analysis, 2008, 46, pp 3032-3070. ⟨10.1137/060666196⟩. ⟨hal-00110436⟩