MS7: Robust discretizations for coupled elliptic/parabolic equations (Part 2)
Chair: Florin Radu
A POSTERIORI ERROR ESTIMATION BY STRESS AND FLUX RECONSTRUCTION FOR BIOT’S CONSOLIDATION MODEL
Abstract: In this talk, flux and stress equilibration procedure for the Biot’s consolidation (see ) problem using Raviart-Thomas elements with weakly symmetry as in [5, 4] are proposed and analyzed. The stress tensor is reconstructed from a displacement-pressure approximation computed with a stable finite element pair. The Darcy velocity is reconstructed in the Raviart-Thomas finite element space, such that both reconstructions are H(div)-conforming, and the equilibration procedure offering serveral advantages (see ) can be used. In particular, these reconstructions are build on vertex patches (see also ) such that they lead to a local efficient a posteriori error estimator for the Biot’s consolidation problem, involving constants that depends only on the shape regularity of the triangulation.
 M. Ainsworth and R. Rankin, Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity, Elsevier, 2005.
 M. A. Biot. General theory of three-dimensional consolidation. J. Appl. Phys., 12:155-169, 1941.
 D. Braess, V. Pillwein and J. Schöberl Equilibrated residual error estimates are p-robust, Comput. Methods Appl. Mech. Engrg 198, 1189?1197.
 F. Bertrand, B. Kober, M. Moldenhauer, G. Starke, Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity , https://arxiv.org/pdf/1808.02655 .pdf
 F. Bertrand, M. Moldenhauer, G. Starke, A posteriori error estimation for planar linear elasticity by stress reconstruction, Computational Methods in Applied Mathematics 2018(4)
Numerical approximation of poromechanic problems with faults and moving boundaries
Daniele Cerroni, Florin Radu, Paolo Zunino
Abstract: Reconstructing the stress and deformation history of a
sedimentary basin is a challenging and
important problem in the geosciences.
Among many other phenomena
erosion plays a key role in the reconstruction of a sedimentary basin history.
To describe erosion, we make the top surface of the basin evolve
according to a given profile.
To account for the presence of faults, the coupling of Stokes/Brinkman and Biot equations is addressed.
We focus on the numerical
approximation of the problem in the
framework of the finite element method.
The poromechanic problem is solved by means of the
fixed stress splitting ,
a sequential procedure where the flow
is solved first followed by the solution of the mechanical problem.
For the computational modeling of faults,
we present and analyze a different coupling approaches for Stokes/Brinkman-Biot coupled equations.
One is based on Nitsche's method  and the other on Lagrange multipliers .
To address the moving boundary, the main challenge is
avoiding re-meshing the computational domain.
We adopt the cut finite element approach (CutFEM) as
described in .
The main issue of this strategy
consists of the ill-conditioning of the
finite element matrices in presence of cut elements of small size.
We show, by means of numerical experiments and theory,
that this issue significantly worsens the performance of
the numerical solver of the discrete problem.
For this reason, we propose a strategy that allows
to overcome the ill-conditioned behavior of
the discrete problem.
The resulting solver is based on the fixed
combined with the ghost penalty
stabilization  and
preconditioning applied to the pressure and
displacement sub-problems, respectively.
Parameter-robust convergence analysis of fixed-stress split iterative scheme for multiple-permeability models
Qingguo Hong, Johannes Kraus, Maria Lymbery, Mary F. Wheeler
Abstract: The fixed-stress split iterative scheme has been widely used to find solutions of problems modelling flow and geomechanics in porous media as this method has been shown to be unconditionally stable. We first present its formulation for flux-based multiple-permeability/multi-porosity poroelasticity systems (MPET equations). Then, we analyse the proposed fixed-stress scheme and prove that it converges linearly with a contraction rate independent of any physical and discretization parameters. Furthermore, numerical test results are presented which support our theoretical findings and clearly demonstrate the advantage of the fixed-stress split iterative scheme over a fully implicit method relying on norm-equivalent preconditioning.
Cost-reducing space-time discretisations for poroelasticity models
Abstract: The quasi-static or dynamic simulation of coupled flow and mechanics appear to be cost-intensive with physics-preserving finite elements. Precisely, the use of high-order discontinuous Galerkin (dG) elements for the quasi-static displacement or dynamic wave fields has shown to be locking-free but costly. To preserve locally the mass for the flow problem, a high-order mixed finite element like the Raviart-Thomas (RT) element can be used. The RT element includes a non-primitive vector-valued element for the flux unknowns and a discontinuous element for the pressure. Such spatial finite elements are expensive in terms of degrees of freedom and the resulting linear systems are typically badly conditioned.
The enriched Galerkin (eG) elements for poroelasticity models are studied recently to reduce the computational costs compared to dG or RT discretisations. The eG elements are composed of a high-order continuous element and a piecewise constant or linear discontinuous enrichment element. The eG discretisation of the mechanics has shown to be almost locking-free and can be proven to be locally mass-preserving for the flow problem. For the discretisation in time a discontinuous, continuous or mixed Galerkin method is applied to resolve the physical problem efficiently. The high-order dG and RT discretisations are compared numerically with cost-reducing high-order enriched Galerkin elements for the flow and mechanics.