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 data, a condition referred to as transverse equilibrium (TVE). To obtain a monotone coarse operator for heterogeneous 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 homogeneous 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 (CMSFV) method 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.

MSC codes

  1. 76S05
  2. 35J99


  1. multiscale
  2. finite volume
  3. multipoint flux approximation
  4. monotonicity
  5. transverse equilibrium
  6. porous media
  7. reservoir simulation
  8. anisotropy

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Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 934 - 962
ISSN (online): 1540-3467


Submitted: 9 October 2007
Accepted: 12 May 2008
Published online: 11 September 2008

MSC codes

  1. 76S05
  2. 35J99


  1. multiscale
  2. finite volume
  3. multipoint flux approximation
  4. monotonicity
  5. transverse equilibrium
  6. porous media
  7. reservoir simulation
  8. anisotropy



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