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An observation on the uniform preconditioners for the mixed Darcy problem
Author(s) -
Bærland Trygve,
Kuchta Miroslav,
Mardal KentAndre,
Thompson Travis
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22500
Subject(s) - schur complement , mathematics , preconditioner , multiphysics , complement (music) , block (permutation group theory) , operator (biology) , darcy's law , mathematical optimization , scaling , intersection (aeronautics) , porous medium , geometry , finite element method , iterative method , porosity , repressor , aerospace engineering , chemistry , engineering , biochemistry , quantum mechanics , transcription factor , thermodynamics , eigenvalues and eigenvectors , physics , geotechnical engineering , complementation , gene , phenotype
Abstract When solving a multiphysics 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 utilized for H (div) ‐based preconditioners. The proposed approach employs an operator preconditioning framework to establish a robust, K ‐uniform block preconditioner. The mapping property of the continuous operator is 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.