Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems
Mathematics of Computation
Enriched multi-point flux approximation for general grids
Journal of Computational Physics
The convergence of mimetic discretization for rough grids
Computers & Mathematics with Applications
A Multipoint Flux Mixed Finite Element Method on Hexahedra
SIAM Journal on Numerical Analysis
Superconvergence of new mixed finite element spaces
Journal of Computational and Applied Mathematics
A co-volume scheme for the rotating shallow water equations on conforming non-orthogonal grids
Journal of Computational Physics
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We present a general framework for constructing and analyzing finite volume methods applied to the mixed formulation of second-order elliptic problems on quadrilateral grids. The control volumes, or covolumes, in the grids overlap. An overlapping finite volume method of this type was first introduced by Russell in [T. F. Russell, Tech. report 3, Reservoir Simulation Research Corp., Tulsa, OK, 1995] and was tested for a variety of problems on rectangular and quadrilateral grids in [Z. Cai et al., Comput Geosci., 1 (1997), pp. 289--315]. Later in [S. H. Chou and D. Y. Kwak, SIAM J. Numer. Anal., 37 (2000), pp. 758--771], Chou and Kwak reformulated it as their mixed covolume method and proved optimal order error estimates using the covolume methodology from [S. H. Chou, Math. Comp., 66 (1997), pp. 85--104] and [S. H. Chou and D. Y. Kwak, SIAM J. Numer. Anal., 35 (1998), pp. 494--507]. However, their treatment was restricted to the case of diagonal coefficient tensor and rectangular grids since a different approach was needed for the quadrilateral (distorted rectangular) case. In this paper we give a new framework, which can handle not only the rectangular anisotropic case but also the anisotropic and irregular grid cases in which the locally supported test functions are images of the natural unit coordinate vectors under the Piola transformation. Our theory sheds light on how to create new test functions using quadratures and now covers Russell's quadrilateral case.