An observation on the uniform preconditioners for the mixed Darcy problem

When solving a multi-physics problem one often decomposes a monolithic system into simpler, frequently single-physics, subproblems. A comprehensive solution strategy may commonly be attempted, then, by means of combining strategies devised for the constituent subproblems. When decomposing the monolithic problem, however, it may be that requiring a particular scaling for one subproblem enforces an undesired scaling on another. In this manuscript we consider the H(div)-based mixed formulation of the Darcy problem as a single-physics subproblem; the hydraulic conductivity, K, is considered intrinsic and not subject to any rescaling. Preconditioners for such porous media flow problems in mixed form are frequently based on H(div) preconditioners rather than the pressure Schur complement. We show that when the hydraulic conductivity, K, is small the pressure Schur complement can also be utilised for H(div)-based preconditioners. The proposed approach employs an operator preconditioning framework to establish a robust, K- and h-uniform block preconditioner. The mapping properties of the continuous operator are a key component in applying the theoretical framework point of view. As such, a main challenge addressed here is establishing a K-uniform inf-sup condition with respect to appropriately weighted Hilbert intersection- and sum-spaces.