Numerical convergence of the MPFA \(O\)-method for general quadrilateral grids in two and three dimensions. (English) Zbl 1110.65106

Aavatsmark, Ivar; Eigestad, Geir Terje; Klausen, Runhild Aae

Arnold, Douglas N. (ed.) et al., Compatible spatial discretizations. Papers presented at IMA hot topics workshop: compatible spatial discretizations for partial differential equations, Minneapolis, MN, USA, May 11–15, 2004. New York, NY: Springer (ISBN 0-387-30916-0/hbk). The IMA Volumes in Mathematics and its Applications 142, 1-21 (2006).

Summary: This paper presents the multipoint flux approximation (MPFA) \(O\)-method for quadrilateral grids, and gives convergence rates for the potential and the normal velocities. The convergence rates are estimated from numerical experiments. If the potential is in \(H^{1+\alpha}\), \(\alpha>0\), the found \(L^2\) convergence order on rough grids in physical space is \(\min\{2,2\alpha\}\) for the potential and \(\min\{1, \alpha\}\) for the normal velocities. For smooth grids the convergence order for the normal velocities increases to \(\min\{2,\alpha\}\). The \(O\)-method is exact for uniform flow on rough grids. This also holds in three dimensions, where the cells may have nonplanar surfaces.
For the entire collection see [Zbl 1097.65003].

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76S05	Flows in porous media; filtration; seepage
35R05	PDEs with low regular coefficients and/or low regular data
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
76T10	Liquid-gas two-phase flows, bubbly flows
76M12	Finite volume methods applied to problems in fluid mechanics

Keywords:

numerical examples; error bounds; stability; control-volume discretization; anisotropy; inhomogeneity; convergence; multipoint flux approximation; numerical experiments; \(O\)-method