The multiscale finite volume (MSFV) method is introduced for the efficient solution of elliptic problems with rough coefficients in the absence of scale separation. The coarse operator of the MSFV method is presented as a multipoint flux approximation (MPFA) with numerical evaluation of the transmissibilities. The monotonicity region of the original MSFV coarse operator has been determined for the homogeneous anisotropic case. For grid-aligned anisotropy the monotonicity of the coarse operator is very limited. A compact coarse operator for the MSFV method is presented that reduces to a 7-point stencil with optimal monotonicity properties in the homogeneous case. For heterogeneous cases the compact coarse operator improves the monotonicity of the MSFV method, especially for anisotropic problems. The compact operator also leads to a coarse linear system much closer to an M-matrix. Gradients in the direction of strong coupling vanish in highly anisotropic elliptic problems with homogeneous Neumann boundary d...