Homogenization: (Joel Dendy and J. David Moulton) In a wide variety of applications, including flow in porous media, conduction in composite materials and neutron transport, the coefficients of the mathematical model exhibit severe variations on two or more significantly different length scales. The accurate coarse scale numerical treatment of these problems relies on a homogenization procedure to define an approximate coarse scale problem that adequately captures the influence of the fine scale structure. Although a multi-scale asymptotic analysis rigorously defines this homogenized problem, it is too costly in general, and thus despite their inherent weaknesses, rudimentary averaging techniques (e.g. arithmetic and harmonic means) are favored in practice. We have developed a new approach motivated by the observation that these some of these multi-scale issues have been resolved in the development of multi-level iterative methods. In particular, the operator induced coarsening of Dendy's robust and efficient black box multigrid must produce coarse grid operators that capture the essential coarse scale influence of the fine scale structure. Using this approach we have developed a local technique that derives approximate homogenized coefficients from the black box coarse grid operators.

Adaptive Grids: (Shengtai Li and Linda Petzold) Adaptive mesh methods can improve the accuracy and efficiency of the numerical approximations to evolutionary and steady state systems of PDEs. These gradients can occur, for example, in boundary layers, shock waves or combustion fronts. To approximate the solution accurately in these regions it is often necessary to generate a mesh that is dense where the solution is rapidly changing. Also, for reasons of efficiency, the mesh should be sparse where the solution is smooth. These adaptive algorithms are costly, but without any local refinement many numerical calculations would be wasteful, or even worse, not resolve some important aspects of the solution satisfactorily. In an evolutionary PDE as the solution changes the mesh must also change to adaptively refine regions where the solution is developing sharp gradients and to remove redundant points from regions where the solution is becoming smoother. Thus, the mesh must have a dynamic behavior much in the same way the solution does. There are illustrate several effective adaptive grid methods and describe some new approaches to implement the methods in a modular efficient way.

Mimetic Finite-Difference Methods for Partial Differential Equation: (Misha Shashkov and Stan Steinberg) The desire to solve new and challenging problems imposes additional requirements on the quality and robustness of numerical algorithms. A number of factors in real problems make this difficult, including non linearity, discontinuity of coefficients and/or solutions, complexity of the physical processes, and 3-D geometry. Experience has shown that the best results usually are obtained by using discrete models that reproduce fundamental properties of the original continuum model of the underlying physical problem. The mimetic finite-difference methods produce algorithms for doing large-scale simulation which are more robust and accurate, because they are based on a solid mathematical theory that emphasizes having the discrete model maintain many of the important properties of the continuum model. The new methodology provides accurate, robust, and stable approximations on non-uniform structured and unstructured grids. Developed during last 15 year in support of the Nuclear Weapons Programs in Los Alamos and Moscow, this approach to construct high-quality finite-difference methods can be used in solving three-dimensional physical and engineering problems. Possible applications involve diffusion processes, modeling fluid flows, electromagnetic. Los Alamos scientists presently are using methods based on mimetic approximations for solution Lagrangian fluid dynamics equation, diffusion equation and flows in porous media.