Computational Geosciences

, Volume 6, Issue 3, pp 285–314

Shape Functions for Velocity Interpolation in General Hexahedral Cells


  • R.L. Naff
    • U.S. Geological Survey
  • T.F. Russell
    • University of Colorado at Denver
  • J.D. Wilson
    • University of Colorado at Denver

DOI: 10.1023/A:1021218525861

Cite this article as:
Naff, R., Russell, T. & Wilson, J. Computational Geosciences (2002) 6: 285. doi:10.1023/A:1021218525861


Numerical methods for grids with irregular cells require discrete shape functions to approximate the distribution of quantities across cells. For control-volume mixed finite-element (CVMFE) methods, vector shape functions approximate velocities and vector test functions enforce a discrete form of Darcy's law. In this paper, a new vector shape function is developed for use with irregular, hexahedral cells (trilinear images of cubes). It interpolates velocities and fluxes quadratically, because as shown here, the usual Piola-transformed shape functions, which interpolate linearly, cannot match uniform flow on general hexahedral cells. Truncation-error estimates for the shape function are demonstrated. CVMFE simulations of uniform and non-uniform flow with irregular meshes show first- and second-order convergence of fluxes in the L2 norm in the presence and absence of singularities, respectively.

control-volume methodCVMFE methoddistorted gridhexahedral gridlocal Darcy lawlocal mass conservationmixed methodPiola transformationvector shape function3-D

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© Kluwer Academic Publishers 2002