Three-points interfacial quadrature for geometrical source terms on nonuniform grids. Application to finite volume schemes for parameter-dependent differential equations.

Theodoros Katsaounis (1) , Chiara Simeoni (2)

1 Department of Mathematics and Applied Mathematics, University of Crete (Heraklion, Greece)
2 DISIM - Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila]

This paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells' size, for which $L^p$-error estimates, $1\le p < +\infty$, are proven. Besides, a suitable notion of (weak) regularity for nonuniform meshes is introduced in the most general case, to compensate possibly reduced consistency conditions, and the optimality of the convergence rates with respect to the regularity assumptions on the problem's data is precisely discussed. This analysis attempts to provide a basic theoretical framework for the numerical simulation on unstructured grids (also generated by adaptive algorithms) of a wide class of mathematical models for real systems (geophysical flows, biological and chemical processes, population dynamics).

Mots clés: nonuniform grids; geometrical source terms; finite volume schemes; optimal convergence rates; consistency

Domaines: Analyse numérique [math.NA]

Theodoros Katsaounis, Chiara Simeoni. Three-points interfacial quadrature for geometrical source terms on nonuniform grids. Application to finite volume schemes for parameter-dependent differential equations.. Calcolo, 2012, 49 (3), pp.149-176. ⟨10.1007/s10092-011-0049-6⟩. ⟨hal-00922835⟩