Multiscale Five-field Composite Mixed Finite Elements for Biot Problems Based on General Polyhedral Partitions
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Biot’s consolidation model is composed of Darcy’s law for the fluid motion, whereas the deformation of the porous media is governed by linear elasticity. We consider a five field mixed method for this problem computing stress and fluid flux as main variables, displacement, pressure and rotation playing the role of Lagrange multipliers associated with divergence and stress symmetry constraints. The methods are designed to cope with complex realistic problems requiring refined computations to capture small structures in the solutions, which usually imply elevated computational costs by standard discretization. As previously explored for Darcy’s flows and for linear elasticity models, the variables are searched in composite finite element spaces based on polyhedral subdomains, using refined discretization inside them, in- stead of coarser normal trace components of tensor and flux taken over subdomain interfaces. We present general error and stability analyses, as well as algorithms for computational implementation of the proposed models, with drastic reduction in the number of degrees-of-freedom.