Shape functions for three-dimensional control-volume mixed
finite-element methods on irregular grids
Richard L. Naff a, Thomas F. Russell b ∗ and John D. Wilson b †
aU.S. Geological Survey, Denver, CO USA
bUniversity of Colorado at Denver, Denver, CO USA
Numerical methods based on unstructured grids, with irregular cells, frequently require
discrete shape functions to approximate the distribution of quantities across cells. For
control-volume mixed finite-element methods, vector shape functions are used to approx-
imate the distribution of velocities across cells. Previous, two-dimensional developments
used linear shape functions to interpolate velocities within a quadrilateral cell. For ir-
regular hexahedral cells in three dimensions, it can be shown that linear shape functions
cannot exactly represent the flux distribution across a cell under uniform flow conditions.
As a result, uniform flow cannot be exactly simulated. A new vector shape function is
proposed for use with irregular hexahedral cells that should provide for a more accu-
rate velocity approximation within a cell. This velocity shape function is a non-linear
interpolator, containing quadratic terms.
1. INTRODUCTION
For simulation of two-dimensional flow in heterogeneous porous media, it has been
shown that mixed methods, and in particular the control-volume mixed finite-element
(CVMFE) methods, are often the most accurate methods for solving for the velocity field
[1,2]. In this paper, we report on a three-dimensional velocity shape functions, based
on covariant vectors for a mapping to a unit cube and for use with irregular hexahedral
cells [3]. The three-dimensional algorithm described herein is based on the CVMFE
methodology as developed by Cai et al. [2] for the simulation of Darcian flow in two
dimensions. In the CVMFE method, the domain is discretized into hexahedral cells that
can have irregular shapes, allowing for the modeling of complex hydrogeological systems.
Shape functions serve as vector basis functions to interpolate the velocity