Efficient Solver for Mixed and Control-Volume Mixed Finite Element Methods in Three Dimensions

John D. Wilson and Thomas F. Russell

Department of Mathematics
University of Colorado at Denver
P.O. Box 173364, Campus Box 170
Denver, CO 80217-3364, U.S.A.


In simulations of groundwater flow in heterogeneous aquifers, accuracy of velocities can be substantially enhanced by discretizations based on mixed or control-volume mixed finite element methods. A potential side benefit is improved accuracy of transport calculations that depend on the flow field. A practical difficulty, especially acute in 3-D, is that these methods lead to discrete linear systems that are not positive definite and cannot be solved by straightforward applications of well-known iterative solvers. We present a specialized iterative solver for mixed methods in 3-D, built around a decomposition of the discrete velocity space that leads to a symmetric positive definite system (the pressure variable is eliminated). The decomposition relies on a local basis for the the divergence-free subspace of the velocity space. The positive definite system is then solved using a preconditioned conjugate gradient method. The unpreconditioned system is ill-conditioned with respect to the dimension of the discrete velocity space, but a preconditioner based on additive or multiplicative domain decomposition makes the condition number independent of this dimension. Numerical results verify uniform convergence of the additive preconditioned conjugate gradient method. We also study numerically the effects of heterogeneity and anisotropy of the conductivity coefficients on the convergence rate. For distorted hexahedral elements, which are trilinear images of cubes that can represent irregular subsurface geology, we consider a control-volume mixed discretization. A similar decomposition of the velocity space is used, hoping to obtain a positive definite system, although the system is not, in general, symmetric. We also present numerical results for the convergence of the preconditioned conjugate gradient method applied to this system.