Applied PDE Group
Nonlinear Partial Differential Equations in Fluid and Solid Mechanics


Three-points interfacial quadrature for geometrical source terms on nonuniform grids

Th. Katsaounis, C. Simeoni, Three-points interfacial quadrature for geometrical source terms on nonuniform gridsCalcolo, 49(3):149-176, 2012
Th. Katsaounis, C. Simeoni
Geometrical source terms, Finite volume schemes, Nonuniform grids, Consistency, Optimal convergence rates
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≤p<+∞, 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).