Domain decomposition for a fractured porous medium

This presentation concerns flow in a porous medium with one or more faults or fractures. In the model to be presented the fractures are treated as interfaces between neighboring subdomains. Then a domain decomposition method, in which the transmission conditions are continuity of the pressure and Darcy flow along the n-1 dimensional fracture interface, is formulated. In an earlier article the model for a domain with a single fracture was defined, existence and uniqueness of the solution were shown, error estimates were obtained and numerical results were shown for a simple two dimensional problem. Here the case of intersecting fractures is treated by imposing continuity of the pressure and continuity of the flux at the intersection of the fractures. This is a more complicated problem both theoretically and from the point of view of implementation. To accelerate the rate of convergence of the iterative procedure a preconditioner is proposed. This preconditioner is based on the observation that in the equation to be solved on the interfaces, the operator corresponding to the Darcy flow along the fracture is of higher order than those corresponding to the Steklov-Poincare operators associated with the neighboring subdomains. Examples and tests evaluating the performance of this preconditioner for a three dimensional model with intersecting fractures are presented. The behavior of the preconditioner with respect to mesh refinement is illustrated. The preconditioner is also compared with a more standard preconditioner.