# Three-Dimensional Control-Volume Mixed Finite Element Methods on Distorted Hexahedral Grids

Monday, January 17, 2000 - 11:00am - 12:15pm

Keller 3-180

Thomas Russell (University of Colorado)

Joint work with Richard L. Naff, USGS, and John D. Wilson, UCD.

The control-volume mixed finite element (CVMFE) method is designed to calculate accurate pressure and velocity distributions in subsurface flow problems with irregular geology and heterogeneous, anisotropic conductivity. Applied to flow equations of prototype - div(K grad p) = f, it differs from the usual lowest-order Raviart-Thomas mixed method in its choice of velocity-vector test functions. On a uniform cartesian grid, the CVMFE velocity test functions are piecewise-constant with respect to control volumes centered around faces of pressure cells. Because cell-centered pressure nodes lie at the ends of these control volumes, the discrete velocity equations can be viewed as representations of Darcy's law on the control volumes with pressures imposed at the ends, in much the same way that discrete continuity equations represent conservation of mass on pressure cells. On distorted hexahedral grids, with possibly heterogeneous, anisotropic K, numerical results show second-order convergence whenever the exact solution is not singular. This talk presents the 3-D formulation in detail; discusses various quadrature formulas, including some that make the discrete system symmetric despite the differences between test and trial functions; presents a sampling of numerical results; and describes an efficient discrete-equation solver currently under development, based on a divergence-free velocity subspace.

The control-volume mixed finite element (CVMFE) method is designed to calculate accurate pressure and velocity distributions in subsurface flow problems with irregular geology and heterogeneous, anisotropic conductivity. Applied to flow equations of prototype - div(K grad p) = f, it differs from the usual lowest-order Raviart-Thomas mixed method in its choice of velocity-vector test functions. On a uniform cartesian grid, the CVMFE velocity test functions are piecewise-constant with respect to control volumes centered around faces of pressure cells. Because cell-centered pressure nodes lie at the ends of these control volumes, the discrete velocity equations can be viewed as representations of Darcy's law on the control volumes with pressures imposed at the ends, in much the same way that discrete continuity equations represent conservation of mass on pressure cells. On distorted hexahedral grids, with possibly heterogeneous, anisotropic K, numerical results show second-order convergence whenever the exact solution is not singular. This talk presents the 3-D formulation in detail; discusses various quadrature formulas, including some that make the discrete system symmetric despite the differences between test and trial functions; presents a sampling of numerical results; and describes an efficient discrete-equation solver currently under development, based on a divergence-free velocity subspace.