  The matched interface and boundary (MIB) method has a proven ability for
delivering the second order accuracy in handling elliptic interface problems
with arbitrarily complex interface geometries. However, its collocation
formulation requires relatively high solution regularity. Finite volume method
(FVM) has its merit in dealing with conservation law problems and its integral
formulation works well with relatively low solution regularity. We propose an
MIB-FVM to take the advantages of both MIB and FVM for solving elliptic
interface problems. We construct the proposed method on Cartesian meshes with
vertex-centered control volumes. A large number of numerical experiments are
designed to validate the present method in both two dimensional (2D) and three
dimensional (3D) domains. It is found that the proposed MIB-FVM achieves the
second order convergence for elliptic interface problems with complex interface
geometries in both $L_{\infty}$ and $L_2$ norms.
