The fixed-stress split method has been widely used as solution method in the coupling of flow and geomechanics. In this work, we analyze the behavior of an inexact version of this algorithm as smoother within a geometric multigrid method, in order to obtain an efficient monolithic solver for Biot's problem. This solver combines the advantages of being a fully coupled method, with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm. Moreover, the fixed-stress split smoother is based on the physics of the problem, and therefore all parameters involved in the relaxation are based on the physical properties of the medium and are given a priori. A local Fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained. The proposed geometric multigrid algorithm is used to solve several tests in semi-structured triangular grids, in order to show the good behavior of the method and its practical utility.

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Keywords Iterative fixed-stress split scheme, Local Fourier analysis, Multigrid, Poroelasticity, Smoother
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Journal Computer Methods in Applied Mechanics and Engineering
Grant This work was funded by the European Commission 7th Framework Programme; grant id h2020/705402 - Efficient numerical methods for deformable porous media. Application to carbon dioxide storage. (poro sos)
Gaspar, F.J, Rodrigo, C, & Schultze, L.M. (2017). On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics. Computer Methods in Applied Mechanics and Engineering, 326, 526–540. doi:10.1016/j.cma.2017.08.025