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

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