[show abstract][hide abstract] ABSTRACT: Numerical methods based on unstructured grids, with irregular cells, usually 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 approximate the distribution of velocities across cells and vector test functions are used to minimize the error associated with the numerical approximation scheme. For a logically cubic mesh, the lowest-order shape functions are chosen in a natural way to conserve intercell fluxes that vary linearly in logical space. Vector test functions, while somewhat restricted by the mapping into the logical reference cube, admit a wider class of possibilities. Ideally, an error minimization procedure to select the test function from an acceptable class of candidates would be the best procedure. Lacking such a procedure, we first investigate the effect of possible test functions on the pressure distribution over the control volume; specifically, we look for test functions that allow for the elimination of intermediate pressures on cell faces. From these results, we select three forms for the test function for use in a control-volume mixed method code and subject them to an error analysis for different forms of grid irregularity; errors are reported in terms of the discrete L2 norm of the velocity error. Of these three forms, one appears to produce optimal results for most forms of grid irregularity.
[show abstract][hide abstract] ABSTRACT: The mixed finite-element approximation to a second-order elliptic PDE results in a saddle-point problem and leads to an indefinite
linear system of equations. The mixed system of equations can be transformed into coupled symmetric positive-definite matrix
equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugate-gradient algorithm is
used for solving the Schur complement problem. The mixed finite-element method is closely related to the cell-centered finite
difference scheme for solving second-order elliptic problems with variable coefficients. For the cell-centered finite difference
scheme, a simple multigrid algorithm can be defined and used as a preconditioner. For distorted grids, an additional iteration
is needed. Nested iteration with a multigrid preconditioned conjugate gradient inner iteration results in an effective numerical
solution technique for the mixed system of linear equations arising from a discretization on distorted grids. Numerical results
show that the preconditioned conjugate-gradient inner iteration is robust with respect to grid size and variability in the
hydraulic conductivity tensor.
[show abstract][hide abstract] 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.