Cell-centered finite difference method for parabolic equation

We develop two cell-centered finite difference schemes for parabolic problems on quadrilateral grids. One scheme is backward Euler scheme with first order accuracy in time increment while the other is Crank–Nicolson scheme with second order accuracy in time increment. 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 velocity elimination, which leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical results indicate first-order convergence in spacial meshsize both for pressures and for subedge flues. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.