Abstract
We develop a cell-centered finite difference method for elliptic problems on curvilinear quadrilateral grids. The method is based on the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element method. A quadrature rule gives a block-diagonal mass matrix and allows for local flux elimination. The method is motivated and closely related to the multipoint flux approximation (MPFA) method. An advantage of our method is that is has a variational formulation. As a result finite element techniques can be employed to analyze the algebraic system and the convergence properites. The method exhibits second order convergence of the scalar variable at the cell-centers and of the flux at the midpoints of the edges. It performs well on problems with rough grids and coefficients, which is illustrated by numerical experiments.
Partially supported by NSF grant DMS 0411413 and the DOE grant DE-FGO2-04ER25617.
Supported in part by the DOE grant DE-FG02-04ER25618, and by the NSF grants DMS 0107389 and DMS 0411694.
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Wheeler, M.F., Yotov, I. (2006). A Cell-Centered Finite Difference Method on Quadrilaterals. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications, vol 142. Springer, New York, NY. https://doi.org/10.1007/0-387-38034-5_10
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