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 a M-matrix.
Gradients in the direction of strong coupling vanish in highly anisotropic elliptic problems with homogeneous Neumann boundary data, a condition referred to as transverse equilibrium (TVE). To obtain a monotone coarse operator for heterogenous problems the local elliptic problems used to determine the transmissibilities must be able to reach TVE as well. This can be achieved by solving two linear local problems with homogenous Neumann boundary conditions, and constructing a third bilinear local problem with Dirichlet boundary data taken from the linear local problems. Linear combination of these local problems gives the MSFV basis functions, but with hybrid boundary conditions that cannot be enforced directly.
The resulting compact multiscale finite volume method (CMSFV) with hybrid local boundary conditions is compared numerically to the original MSFV method. For isotropic problems both methods have comparable accuracy, but the CMSFV method is robust for highly anisotropic problems where the original MSFV method leads to unphysical oscillations in the coarse solution and recirculations in the reconstructed velocity field.