  The finite volume methods are frequently employed in the discretization of
diffusion problems with interface. In this paper, we firstly present a
vertex-centered MACH-like finite volume method for solving stationary diffusion
problems with strong discontinuity and multiple material cells on the Eulerian
grids. This method is motivated by Frese [No. AMRC-R-874, Mission Research
Corp., Albuquerque, NM, 1987]. Then, the local truncation error and global
error estimates of the degenerate five-point MACH-like scheme are derived by
introducing some new techniques. Especially under some assumptions, we prove
that this scheme can reach the asymptotic optimal error estimate $O(h^2 |\ln
h|)$ in the maximum norm. Finally, numerical experiments verify theoretical
results.
