Convergence of a symmetric MPFA method on quadrilateral grids

This paper investigates different variants of the multipoint flux approximation (MPFA) O-method in 2D, which rely on a transformation to an orthogonal reference space. This approach yields a system of equations with a symmetric matrix of coefficients. Different methods appear, depending on where the transformed permeability is evaluated. Midpoint and corner-point Wheeler was partially supported by NSF grant DMS 0411413 and the DOE grant DE-FGO2-04ER25617. Yotov was supported in part by the DOE grant DE-FG02-04ER25618, the NSF grant DMS 0411694 and the J. Tinsley Oden Faculty Fellowship, The University of Texas at Austin. I. Aavatsmark (B) · G. T. Eigestad · R. A. Klausen Centre for Integrated Petroleum Research, University of Bergen, P.O. Box 7800, 5020 Bergen, Norway e-mail: [email protected] G. T. Eigestad e-mail: [email protected] R. A. Klausen Centre of Mathematics for Applications, University of Oslo, Mail: CMA, P.O. Box 1053 Blindern, 0316 Oslo, Norway e-mail: [email protected] M. F. Wheeler Institute for Computational Engineering and Sciences (ICES), Department of Aerospace Engineering & Engineering Mechanics, and Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712, USA e-mail: [email protected] I. Yotov Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260, USA e-mail: [email protected] evaluations are considered. Relations to mixed finite element (MFE) methods with different velocity finite element spaces are further discussed. Convergence of the MPFA methods is investigated numerically. For corner-point evaluation of the reference permeability, the same convergence behavior as the O-method in the physical space is achieved when the grids are refined uniformly or when grid perturbations of order h2 are allowed. For h2-perturbed grids, the convergence of the normal velocities is slower for the midpoint evaluation than for the corner-point evaluation. However, for rough grids, i.e., grids with perturbations of order h, contrary to the physical space method, convergence cannot be claimed for any of the investigated reference space methods. The relations to the MFE methods are used to explain the loss of convergence.