In this paper, we firstly present a vertex-centered MACH-like finite volume method for the diffusion problems with variable and jump coefficients on rectangular grids. Then, its corresponding auxiliary diffusion problem with constant and jump coefficients is introduced, and the inverse of the coefficient matrix of the auxiliary problem's discrete system is used as the preconditioner. By using the special algebraic structure of the inverse matrix and together with a dimensionality reduction technique and the fast fourier transform, a direct method with low computational complexity (O(N ln(N))) is designed to achieve the behavior of our preconditioner. Numerical results show that the PCG algorithm with this preconditioner is efficient and robust.
. AN EFFICIENT PRECONDITIONER FOR FINITE VOLUME APPROXIMATION OF DIFFUSION PROBLEMS WITH VARIABLE AND JUMP COEFFICIENTS[J]. Journal of Numerical Methods and Computer Applicat, 2018, 39(2): 150-160.
Ewing R E, Iliev O P, Lazarov R D. A Modified Finite Volume Approximation Of Second-Order Elliptic Equations With Discontinuous Coefficients[J]. Siam Journal on Scientific Computing, 2001, 23(4):1335-1351.
Luce R, Perez S. A finite volume scheme for an elliptic equation with heterogeneous coefficients. Application to a homogenization problem[J]. Applied Numerical Mathematics, 2001, 38(4):427- 444.
Oevermann M, Klein R. A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces[J]. Journal of Computational Physics, 2006, 219(2):749-769.
Latige M, Colin T, Gallice G. A second order Cartesian finite volume method for elliptic interface and embedded Dirichlet problems[J]. Computers & Fluids, 2013, 83(11):70-76.
Frese M H. MACH2:A two-dimensional magnetohydrodynamic simulation code for complex experimental configurations[J]. Interim Report Mission Research Corp Albuquerque Nm, 1987.
Peng J, Shu S, Yu H, et al. Error estimates on a finite volume method for diffusion problems with interface on rectangular grids[J]. Applied Mathematics and Computation, 2017, 311:335-352.
Shumlak U, Hussey T W, Peterkin R E. Mach3:A Three-dimensional MHD Code[C]. Plasma Science, 1994. Conference Record-Abstracts., 1994 IEEE International Conference on. IEEE, 1994:75-75.