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When solving a multi-physics problem one often decomposes a monolithic systeminto simpler, frequently single-physics, subproblems. A comprehensive solutionstrategy may commonly be attempted, then, by means of combining strategiesdevised for the constituent subproblems. When decomposing the monolithicproblem, however, it may be that requiring a particular scaling for onesubproblem enforces an undesired scaling on another. In this manuscript weconsider the H(div)-based mixed formulation of the Darcy problem as asingle-physics subproblem; the hydraulic conductivity, K, is consideredintrinsic and not subject to any rescaling.  Preconditioners for such porous media flow problems in mixed form arefrequently based on H(div) preconditioners rather than the pressure Schurcomplement. We show that when the hydraulic conductivity, K, is small thepressure Schur complement can also be utilised for H(div)-basedpreconditioners. The proposed approach employs an operator preconditioningframework to establish a robust, K- and h-uniform block preconditioner. Themapping properties of the continuous operator are a key component in applyingthe theoretical framework point of view. As such, a main challenge addressedhere is establishing a K-uniform inf-sup condition with respect toappropriately weighted Hilbert intersection- and sum-spaces.

An Observation On The Uniform Preconditioners For The Mixed Darcy Problem