FINITE VOLUME SCHEMES FOR DIFFUSION EQUATIONS: INTRODUCTION TO AND REVIEW OF MODERN METHODS

Jerome Droniou (1)

1 School of Mathematical Sciences [Clayton]

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature results, focusing on two important properties of schemes (discrete versions of well-known properties of the continuous equation): coercivity and minimum-maximum principles. Coercivity ensures the stability of the method as well as its convergence under assumptions compatible with real-world applications, whereas minimum-maximum principles are crucial in case of strong anisotropy to obtain physically meaningful approximate solutions.

Mots clés: review; elliptic equation; finite volume schemes; multi-point flux approximation; hybrid mimetic mixed methods; discrete duality finite volume schemes; coercivity; convergence analysis; monotony; minimum and maximum principles

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

Jerome Droniou. FINITE VOLUME SCHEMES FOR DIFFUSION EQUATIONS: INTRODUCTION TO AND REVIEW OF MODERN METHODS. Mathematical Models and Methods in Applied Sciences, 2014, 24 (8), pp.1575-1619. ⟨10.1142/S0218202514400041⟩. ⟨hal-00813613v3⟩