Robust Approximation of Generalized Biot-Brinkman Problems

Abstract

The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.

Appendix A. Components of Multigrid Preconditioner

In this section we report numerical experiments demonstrating robustness of geometric multigrid preconditioners for blocks $\mathcal {B}_{\varvec{u}}$ and $\mathcal {B}_{\varvec{v}}$ of the Biot-Brinkman preconditioner (4.12). Adapting the unit square geometry and the setup of boundary conditions from Sect. 5.3 we investigate performance of the preconditioners by considering boundedness of the (preconditioned) conjugate gradient (CG) iterations. In the following, the initial vector is set to 0 and the convergence of the CG solver is determined by reduction of the preconditioned residual norm by a factor $10^{8}$. Finally, both systems are discretized by $\text {BDM}_1$ elements.

Table 2 confirms robustness of the F(2, 2)-cycle for the displacement block of (4.12). In particular, the iterations can be seen to be bounded in mesh size and the Lamé parameter $\lambda$.

Table 2 Number of preconditioned conjugate gradient iterations for approximating the displacement block $\mathcal {B}_{\varvec{u}}$ of the Biot-Brinkman preconditioner

For the flux block $\mathcal {B}_{\varvec{v}}$ we limit the investigations to the two-network case and set $c_2=0$, $\alpha _2=1$ as these parameter values yielded the stiffest problems (in terms of their condition numbers) in the robustness study of Sect. 5.2. Performance of the geometric multigrid preconditioner using a W(2, 2)-cycle with vertex-star smoother is then summarized in Fig. 11. We observe that the number of CG iterations is bounded in the mesh size and variations in $K_2$, $\nu _2$ and the exchange coefficient $\beta$. We remark that for some parameter configurations the observed dependence of the iteration counts is not monotone in mesh size. In particular, the number of preconditioned CG iterations on a finer mesh can be smaller than on a coarse one. However, in these cases the difference is 1 or 2 iterations with the former being the typical value.

Fig. 11

Number of preconditioned conjugate gradient iterations for approximating the flux block $\mathcal {B}_{\varvec{v}}$ of the Biot-Brinkman preconditioner. The preconditioner uses W(2, 2)-cycle of geometric multigrid with vertex-star (damped Richardson) smoother and 3 grid levels. (Top) Transfer coefficient $\beta =10^{6}$, (bottom) $\beta =10^{-6}$. Values of $K_2$, $\nu _2$ (encoded by markers) and $\lambda$ (encoded by line color) are varied. In both setups $c_2=0$, $\alpha _2=1$ and the remaining problem parameters are set to 1