The mimetic finite difference method on polygonal meshes for diffusion-type problems

T. Arbogast, L. Cowsar, M. Wheeler and I. Yotov, Mixed finite element methods on non-matching multiblock grids, SIAM J. Numer. Anal. 37 (2000) 1295–1315.

M. Berndt, K. Lipnikov, D. Moulton and M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation, J. Numer. Math. 9 (2001) 265–284.

M. Berndt, K. Lipnikov, M. Shashkov, M. Wheeler and I. Yotov, Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals, Los Alamos Report LA-UR-03-7904 (submitted to SIAM J. Numer. Anal.).

J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys. 172 (2001) 739–765.

R.E. Ewing, R.D. Lazarov and P.S. Vassilevski, Local refinement techniques for elliptic problems on cell-centered grids I, error analysis, Math. Comp. 56 (1991) 437–461.

R.E. Ewing, M. Liu and J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals, SIAM J. Numer. Anal. 36(3) (1999) 772–787.

J. Hyman, J. Morel, M. Shashkov and S. Steinberg, Mimetic finite difference methods for diffusion equations, Comput. Geosci. 6(3/4) (2002) 333–352.

J. Hyman and M. Shashkov, Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion, Progress Electromagnetic Res. 32 (2001) 89–121.

J. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, J. Comput. Phys. 132 (1997) 130–148.

J. Hyman, M. Shashkov and S. Steinberg, The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods, Internat. J. Comput. Math. Appl. 42 (2001) 1527–1547.

Y. Kuznetsov, Two-level preconditioners with projectors for unstructured grids, Russian J. Numer. Anal. Math. Modelling 15(3/4) (2000) 247–256.

Y. Kuznetsov and S. Repin, New mixed finite element method on polygonal and polyhedral meshes, Russian J. Numer. Anal. Math. Modelling 18(3) (2003) 261–278.

K. Lipnikov, J. Morel and M. Shashkov, Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes, J. Comput. Phys. 199 (2004) 589–597.

R. Liska, M. Shashkov and V. Ganzha, Analysis and optimization of inner products for mimetic finite difference method on triangular grid, Math. Comput. Simulation 67 (2004) 55–66.

L. Margolin, M. Shashkov and P. Smolarkiewicz, A discrete operator calculus for finite difference approximations, Comput. Methods Appl. Mech. Engrg. 187 (2000) 365–383.

J. Morel, R. Roberts and M. Shashkov, A local support-operators diffusion discretization scheme for quadrilateral r−z meshes, J. Comput. Phys. 144 (1998) 17–51.

R. Naff, T. Russell and J. Wilson, Shape functions for velocity interpolation in general hexahedral cells, Comput. Geosci. 6(3/4) (2002) 285–314.

A. Okabe, B. Boots, K. Sugihara and S.N. Chiu, Concepts and Applications of Voronoi Diagrams (Wiley, New York, 2000).

T.S. Palmer, Discretizing the diffusion equation on unstructured polygonal meshes in two dimensions, Ann. Nuclear Energy 28 (2001) 1851–1880.

M. Shashkov, Conservative Finite-Difference Methods on General Grids (CRC Press, Boca Raton, FL, 1996).

M. Shashkov and S. Steinberg, Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys. 129 (1996) 383–405.

K. Stüben, Algebraic multigrid (AMG): Experiences and comparisons, Appl. Math. Comput. 13 (1983) 419–452.

P. Vanêk, M. Brezina and J. Mandel, Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math. 88 (2001) 559–579.