A compact multipoint flux approximation method with improved robustness

Multipoint flux approximation (MPFA) methods were introduced to solve control‐volume formulations on general grids. Although these methods are general in the sense that they may be applied to any grid, their convergence and monotonicity properties vary. We introduce a new MPFA method for quadrilateral grids termed the L‐method. This method seeks to minimize the number of entries in the flux stencils, while honoring uniform flow fields. The methodology is valid for general media. For homogeneous media and uniform grids in two dimensions, this method has four‐point flux stencils and seven‐point cell stencils, whereas the MPFA O‐methods have six‐point flux stencils and nine‐point cell stencils. The reduced stencil of the L‐method appears as a consequence of adapting the method to the closest neighboring cells, or equivalently, to the dominating principal direction of anisotropy. We have tested the convergence and monotonicity properties for this method and compared it with the O‐methods. For moderate grids, the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the O‐methods is lost, while the L‐method converges with a reduced convergence rate. Also, the L‐method has a larger monotonicity range than the O‐methods. For homogeneous media and uniform parallelogram grids, the matrix of coefficients is an M‐matrix whenever the method is monotone. For strongly nonmonotone cases, the oscillations observed for the O‐methods are almost removed for the L‐method. Instead, extrema on no‐flow boundaries are observed. These undesired solutions, which only occur for parameters not common in applications, should be avoided by requiring that the previously derived monotonicity conditions are satisfied. For local grid refinements, test runs indicate that the L‐method yields almost optimal solutions, and that the solution is considerably better than the solutions obtained by the O‐methods. The efficiency of the linear solver is in many cases better for the L‐method than for the O‐methods. This is due to lower condition number and a reduced number of entries in the matrix of coefficients. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008