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The principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought. However, in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost, or obeyed only to an approximation that… (More)

- Franco Brezzi, Konstantin Lipnikov, Mikhail J. Shashkov
- SIAM J. Numerical Analysis
- 2005

The stability and convergence properties of the mimetic finite difference method for diffusion-type problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved.

operators, in this case, the divergence =? and the gradient =: A finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is constructed for logically rectangular grids. The performance of this algorithm is comparable to other algorithms for problems with a u t 5 = · K =u 1 f. (1.1)… (More)

- Hyung Taek Ahn, Mikhail J. Shashkov
- J. Comput. Physics
- 2007

We describe multi-material (more than two materials) interface reconstruction methods for 3D mesh of generalized polyhedrons. Basic information used in interface reconstruction is volume fractions of each material in mixed cell containing several materials. All methods subdivide mixed cell into set of pure non-overlapping sub-cells each containing just one… (More)

- J. Hyman, M. Shashkov
- 2002

This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and… (More)

In this paper we present a new formulation of the artificial viscosity concept. Physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated. The first important property is that by definition a viscosity must always be dissipative, transferring… (More)

A new second-order finite-difference algorithm for the numerical where nW is the vector of unit outward normal to the boundsolution of diffusion problems in strongly heterogeneous and nonary V, and a and c are functions given on V. The algoisotropic media is constructed. On problems with rough coefficients or highly nonuniform grids, the new algorithm is… (More)

The main goal of this paper is to establish the convergence of mimetic discretizations of the firstorder system that describes linear stationary diffusion on general polyhedral meshes. The main idea of the mimetic finite difference (MFD) method is to mimic the underlying properties of the original continuum differential operators, e.g. conservation laws,… (More)

A procedure is presented to improve the quality of surface meshes while maintaining the essential characteristics of the discrete surface. The surface characteristics are preserved by repositioning mesh vertices in a series of element-based local parametric spaces such that the vertices remain on the original discrete surface. The movement of the mesh… (More)

The bane of Lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a Lagrangian zonal mass has ceased to be valid. At short spatial grid scales this is usually referred to by the terms “hourglass” mode or “keystone” motion… (More)