CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   Control-Volume Mixed FEMs and Efficient Solvers
         for Heterogeneous Groundwater Flow Equations

 

SPEAKER: Thomas F. Russell, Department of Mathematics, 
         University of Colorado at Denver

         
DATE:    Monday, November 3, 1997  


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


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



ABSTRACT

The ability to compute accurate velocities is important for
applications of groundwater flow and transport codes.  In highly
heterogeneous porous media, this is difficult for standard
numerical methods.  Irregular geological features, which
suggest the use of irregular grids, and variable directions
of anisotropy add to the challenge.  Mixed finite element
methods can overcome these obstacles and
produce accurate results.  However, the discrete
formulations are often complex, and the linear algebraic
equations are not amenable to standard solvers and are
difficult to solve as efficiently as equations from other
methods.  This has inhibited practical applications,
especially in 3D.

A control-volume variant of the lowest-order Raviart-Thomas
mixed method is presented for general logically rectangular
grids (2D quadrilaterals, 3D hexalaterals).  The Darcy
equation is enforced on a cell-sized ``tank'' (control
volume) around each degree of freedom of the velocity (edge
in 2D, face in 3D).  The discrete equations are
simple, easy to implement, and involve only cell pressures
and edge or face fluxes (integrated normal velocities).
Numerical tests in 2D show second-order convergence of the
edge fluxes whenever the exact solution is not singular.
The accuracy far exceeds that of commonly used methods.

Also presented is an efficient solver for the 3D equations.
It uses a convenient basis for the divergence-free
velocity trial functions to reduce the equations to a
symmetric positive-definite system of smaller size than has
been possible previously.  This enables the 3D equations to
be solved with effort comparable to that for other methods.