UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 626 UCD Building, 1250 14th St., Denver

TIME: noon (Refreshments served at 11:45 am)

DATE: Monday, September 28, 1998

The Numerical Solution of Diffusion Problems
in Strongly Heterogeneous Non-Isotropic Materials

      James Hyman, Mikhail Shashkov

Los Alamos National Laboratory, T-7, MS-B284
       Los Alamos NM 87545
    http://cnls.lanl.gov/~shashkov

                 and

           Stanly Steinberg

   Department of Mathematics and Statistics
University of New Mexico, Albuquerque NM 87131

               ABSTRACT

A new second-order finite-difference algorithm for the numerical
solution of diffusion problems in strongly heterogeneous and
non-isotropic media is constructed. On problems with rough coefficients
or highly nonuniform grids, the new algorithm is superior to all
other algorithms we have compared it with.
For problems with smooth coefficients on smooth grids, the method
is comparable with other second-order methods. The new algorithm
is formulated for logically-rectangular
grids and is derived using the support-operators method.

A key idea in deriving the method was to replace the usual inner product
of vector functions by an inner product weighted by the inverse of the
material properties tensor and to use the flux operator,
defined as the material properties tensor times the
gradient, rather than the gradient, as one of the basic first-order operators
in the support-operators method. The discrete analog of the flux operator
must also be the negative adjoint of the discrete divergence,
in an inner product that is a discrete analog of the continuum inner
product. The resulting method is conservative and the
discrete analog of variable coefficient Laplacian is symmetric and negative
definite on nonuniform grids. In addition, on any grid, the discrete divergence
is zero on constant vectors, the null space for the gradient are the constant
functions and, when the material properties are piecewise constant,
the discrete flux operator is exact for piece-wise linear
functions. We also will discuss how to increase accuracy of the fluxes
by modifying the inner product in the space of discrete vector functions.
  We compare the methods on some of the most difficult examples to be found
in the literature.