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Comparison of monolithic and splitting solution schemes for dynamic porous media problems
Author(s) -
Markert B.,
Heider Y.,
Ehlers W.
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2789
Subject(s) - discretization , finite element method , mathematics , poromechanics , partial differential equation , porous medium , algebraic equation , mathematical analysis , algebraic number , nonlinear system , physics , engineering , porosity , geotechnical engineering , quantum mechanics , thermodynamics
Proceeding from the governing equations describing a saturated poroelastic material with intrinsically incompressible solid and fluid constituents, we compare the monolithic and splitting solution of the different multi‐field formulations feasible in porous media dynamics. Because of the inherent solid–fluid momentum interactions, one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint. Here, the resulting set of differential‐algebraic equations (DAE) is solved by the finite element method (FEM) following two different strategies: (1) an implicit monolithic approach, where the equations are first discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators; (2) a semi‐explicit–implicit splitting scheme in the sense of a fractional‐step method, where the DAE are first discretized in time, split using intermediate variables, and then discretized in space using linear equal‐order approximations for all primary unknowns. Finally, a one‐ and a two‐dimensional wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies. Therefore, several test cases differing in the used multi‐field formulation, the monolithic time‐stepping method, and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter. In the end, we provide a reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem. Copyright © 2009 John Wiley & Sons, Ltd.