Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition

Boris Andreïanov (1) , Mohamed Karimou Gazibo (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

This note is devoted to the study of the finite volume methods used in the discretization of degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The notion of entropy-process solution, successfully used for the Dirichlet problem, is insufficient to obtain a uniqueness and convergence result because of a lack of regularity of solutions on the boundary. We infer the uniqueness of entropy-process solution using the tool of the nonlinear semigroup theory by passing to the new abstract notion of integral-process solution. Then, we prove that numerical solution converges to the unique entropy solution as the mesh size tends to 0.

Mots clés: Degenerate parabolic problem; Stationary problem; Entropy solution; Integral solution; Nonlinear semigroup theory; Finite volume scheme

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreïanov, Mohamed Karimou Gazibo. Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition. 2014. ⟨hal-00950142v1⟩