A cell-centered finite difference method on quadrilaterals. (English) Zbl 1110.65099

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, 189-207 (2006).
Summary: 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 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 method. An advantage of our method is that it has a variational formulation.
As a result finite element techniques can be employed to analyze the algebraic system and the convergence properties. 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.
For the entire collection see [Zbl 1097.65003].


65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs