Multipoint MFEM for Biot poroelasticity

Published version ArXiv (open access)

Key ideas

• By rewriting the linear elasticity equations as a weighted vector-Laplacian, we obtain a four-field formulation of Biot poroelasticity in terms of solid rotation, solid displacement, fluid flux, and fluid pressure.
• The relevant function spaces form a differential complex that can be preserved by discretizing with lowest order finite element spaces.
• A structure-preserving quadrature rule allows us to eliminate the solid rotation and fluid flux.
• The system is analyzed using parameter-weighted norms and perturbed saddle point theory.

The finite element spaces for solid rotation, solid displacement, fluid flux, and fluid pressure. Both the solid rotation and fluid flux are eliminated locally, resulting in a scheme with one degree of freedom per facet and cell.

Main findings

• The resulting method uses employs the lowest order Raviart-Thomas finite element pair for the displacement and pressure variables.
• Linear convergence in all variables is shown both theoretically and experimentally.
• Certain properties of the solutions remain invariant under the low-order quadrature, including the curl of the rotation.
• The analysis in parameter-weighted norms guides the construction of preconditioners that are robust in all relevant physical limits.
• A drawback is that the formulation is based on unconventional boundary conditions and currently does not directly handle traction conditions.