The discretization of subsurface fluid flow and transport equations using the finite element method leads to a system of nonlinear equations that must be solved at every time step. The nonlinear equations are solved using Newton's method, which requires that the Newton step be calculated. The Newton step is calculated as the solution of a linear system, and inexact Newton methods typically use a linear iterative method to solve for the Newton step.; These ideas have been put into practice in the A&barbelow;daptive H&barbelow;ydrology model, a production code written by employees of the Army Corps of Engineers Engineer Research and Development Center located in Vicksburg, Mississippi. The ADH model simulates three dimensional flow and transport using tetrahedral elements in space. The iterative linear methods that are used to solve for the Newton step are Krylov subspace methods, and the performance of these methods is improved with the use of a preconditioner. This work was based on finding effective preconditioners that would work well in serial and in parallel. The main contribution to ADH was in the implementation of two-level preconditioners based on domain decomposition preconditioning strategies. Both one- and two-level preconditioners are presented here, along with a discussion on the development of the second level problem. The theoretical bound on the condition number of the two-level preconditioned system and numerical verification of the theory are provided. Numerical results demonstrate the effectiveness of the two-level preconditioner on a nonlinear scalar equation and on a system of nonlinear equations, both of which are from the hydrology literature.
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