Abstract
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 L 2 norm in the presence and absence of singularities, respectively.
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Naff, R., Russell, T. & Wilson, J. Shape Functions for Velocity Interpolation in General Hexahedral Cells. Computational Geosciences 6, 285–314 (2002). https://doi.org/10.1023/A:1021218525861
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DOI: https://doi.org/10.1023/A:1021218525861