Publications (5)2.84 Total impact

Article: Test Functions for ThreeDimensional ControlVolume Mixed FiniteElement Methods on Irregular Grids
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ABSTRACT: Numerical methods based on unstructured grids, with irregular cells, usually require discrete shape functions to approximate the distribution of quantities across cells. For controlvolume mixed finiteelement 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 lowestorder 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 controlvolume 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.11/2012;  [Show abstract] [Hide abstract]
ABSTRACT: The mixed finiteelement approximation to a secondorder elliptic PDE results in a saddlepoint problem and leads to an indefinite linear system of equations. The mixed system of equations can be transformed into coupled symmetric positivedefinite matrix equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugategradient algorithm is used for solving the Schur complement problem. The mixed finiteelement method is closely related to the cellcentered finite difference scheme for solving secondorder elliptic problems with variable coefficients. For the cellcentered 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 conjugategradient inner iteration is robust with respect to grid size and variability in the hydraulic conductivity tensor.Computational Geosciences 01/2010; 14(2):289299. · 1.42 Impact Factor  [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 controlvolume mixed finiteelement (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 Piolatransformed shape functions, which interpolate linearly, cannot match uniform flow on general hexahedral cells. Truncationerror estimates for the shape function are demonstrated. CVMFE simulations of uniform and nonuniform flow with irregular meshes show first and secondorder convergence of fluxes in the L 2 norm in the presence and absence of singularities, respectively.Computational Geosciences 01/2002; 6(3):285314. · 1.42 Impact Factor 
Article: Shape functions for threedimensional controlvolume mixed finiteelement methods on irregular grids
01/2002; 

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35  Citations  
2.84  Total Impact Points  
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2002–2010

United States Geological Survey
Reston, Virginia, United States
