Parameter-robust preconditioner for Stokes-Darcy coupled problem with Lagrange multiplier

TL;DR

The paper tackles a parameter-robust solver for a coupled Stokes-Darcy system enforced via a Lagrange multiplier at the interface. It develops an operator-preconditioning framework yielding a block preconditioner whose performance is uniform across physical and discretization parameters in most boundary configurations, while identifying near-kernel scenarios that degrade typical methods. A MINRES convergence analysis reveals stagnation due to slow eigenvalues, which is mitigated by a deflation strategy targeting the near-kernel subspace. Numerical experiments confirm robustness and demonstrate acceleration of convergence, including in complex settings like floating Darcy subdomains.

Abstract

In this paper, we propose a parameter-robust preconditioner for the coupled Stokes-Darcy problem equipped with various boundary conditions, enforcing the mass conservation at the interface via a Lagrange multiplier. We rigorously establish that the coupled system is well-posed with respect to physical parameters and mesh size and provide a framework for constructing parameter-robust preconditioners. Furthermore, we analyze the convergence behavior of the Minimal Residual method in the presence of small outlier eigenvalues linked to specific boundary conditions, which can lead to slow convergence or stagnation. To address this issue, we employ deflation techniques to accelerate the convergence. Finally, Numerical experiments confirm the effectiveness and robustness of the proposed approach.

Figures (12)

• Figure 1: Different configurations of boundary conditions considered with the coupled Stokes-Darcy problem. (a)-(d) do not lead to near kernel. Highlighted in color is the change of boundary conditions in (c) and (d), which yields a near kernel in cases (e) and (f).
• Figure 1: Convergence history of the MINRES method (plotted against the left vertical-axis), the harmonic Ritz value, and the term $F_k$ (both plotted against the right vertical axis) for the NE case when $\alpha_{\text{BJS}} = 0.5$ and $h = h_0/4$.
• Figure 2: Performance of preconditioner \ref{['preconditioner']} for the Stokes-Darcy problem, applied to the geometry from \ref{['ex:mms']}, under different boundary condition configurations (see also \ref{['fig:bc_config']}) and parameter variations. We set $\alpha_{\text{BJS}}=0.5$. Defining the multiplier preconditioner $S$ as in \ref{['eq:interface_preconditioner']} yields to robustness in $\mu$ and $K$ as the considered boundary condition configurations do not lead to near-kernel.
• Figure 2: Convergence history of the MINRES method (plotted against the left vertical axis), the harmonic Ritz value, and the term $F_k$ (plotted against the right vertical axis) for the EN case with $\alpha_{\text{BJS}} = 0.5$ and $h = h_0/4$.
• Figure 3: Performance of preconditioner \ref{['preconditioner']} for the Stokes-Darcy problem, applied to the geometry from \ref{['ex:mms']}, with different parameter values for the EN and NE cases (see also \ref{['fig:bc_config']}). Condition numbers are sensitive to the product $\mu K$ due to the present near-kernel.

Theorems & Definitions (21)

• Theorem 2.1: Well-posedness of \ref{['abstract-coupled-equ']}
• Proof 1
• Lemma 3.1: Stokes subproblem with mixed boundary conditions
• Proof 2
• Remark 3.2: Stokes subproblem with pure Dirichlet conditions
• Lemma 3.3: Darcy subproblem with mixed boundary conditions
• Proof 3
• Remark 3.4: Darcy subproblem with pure Dirichlet conditions
• Theorem 3.5
• Theorem 3.6