[Paper Review] Robust preconditioning of monolithically coupled multiphysics problems

Karl Erik Holter, Miroslav Kuchta+1 more|arXiv (Cornell University)|Jan 15, 2020

Advanced Numerical Methods in Computational Mathematics53 references6 citations

TL;DR

This paper presents a robust, parameter-robust preconditioning framework for monolithically coupled multiphysics problems using fractional and weighted Laplacian operators at the interface. By leveraging operator preconditioning in fractional Sobolev spaces, the method achieves mesh-independent convergence for challenging problems like Darcy-Stokes and fluid-structure interaction, with numerical experiments confirming optimal convergence rates and bounded condition numbers across varying material parameters and mesh refinements.

ABSTRACT

In many applications, one wants to model physical systems consisting of two different physical processes in two different domains that are coupled across a common interface. A crucial challenge is then that the solutions of the two different domains often depend critically on the interaction at the interface and therefore the problem cannot be easily decoupled into its subproblems. Here, we present a framework for finding robust preconditioners for a fairly general class of such problems by exploiting operators representing fractional and weighted Laplacians at the interface. Furthermore, we show feasibility of the framework for two common multiphysics problems; namely the Darcy-Stokes problem and a fluid--structure interaction problem. Numerical experiments that demonstrate the effectiveness of the approach are included.

Motivation & Objective

• To develop a robust preconditioning strategy for monolithic formulations of coupled multiphysics problems where subproblems are tightly coupled at an interface.
• To ensure preconditioners are parameter-robust with respect to variations in physical parameters such as viscosity, permeability, and the Beavers-Joseph-Saffman coefficient.
• To extend operator preconditioning theory to interface-coupled problems using weighted and fractional Sobolev spaces.
• To validate the framework numerically on two canonical problems: Darcy-Stokes flow and fluid-structure interaction.
• To demonstrate that the preconditioned system maintains bounded condition numbers and optimal convergence rates under mesh refinement.

Proposed method

• Formulate the coupled multiphysics problem using a monolithic weak form with Lagrange multipliers to enforce interface continuity.
• Construct preconditioners based on Riesz mappings in fractional and weighted Sobolev spaces to capture the correct scaling of interface variables.
• Use fractional Laplacian operators at the interface to model the transmission conditions, including the Beavers-Joseph-Saffman condition.
• Derive norms for velocity, pressure, and Lagrange multiplier that reflect the physical scaling and ensure well-posedness of the coupled system.
• Apply the preconditioner within a MINRES iterative solver to achieve mesh-convergent and parameter-robust convergence.
• Validate the method using numerical experiments on 2D Darcy-Stokes and fluid-structure interaction problems with varying mesh sizes and material parameters.

Experimental results

Research questions

• RQ1 Can a monolithic preconditioning framework be constructed that remains robust with respect to variations in physical parameters such as viscosity and permeability in coupled multiphysics problems?
• RQ2 How can fractional and weighted Sobolev spaces be used to design preconditioners that reflect the correct scaling of interface variables in coupled systems?
• RQ3 Does the proposed preconditioner maintain bounded condition numbers and optimal convergence rates under mesh refinement for problems like Darcy-Stokes and fluid-structure interaction?
• RQ4 What is the relative performance of the monolithic preconditioner compared to domain decomposition methods in terms of solution time and iteration count?
• RQ5 Can the framework be applied to problems where the well-posedness of the subproblems is not classically established, under suitable assumptions?

Key findings

• The proposed preconditioner achieves bounded condition numbers (e.g., 12.72 at h = 2⁻⁸ for the Stokes-Navier problem) and optimal convergence rates across mesh refinements.
• Numerical experiments show that the monolithic preconditioner outperforms domain decomposition methods, with solution times of ~47s per iteration for Darcy-Stokes and ~116s for Stokes-Navier, compared to 60–150s for DD with similar performance.
• The method maintains robustness across parameter variations, including extreme values of the Beavers-Joseph coefficient (k = 10⁻⁶ to 1), with stable convergence.
• The convergence of the MINRES solver is mesh-independent and robust with respect to physical parameters, as evidenced by consistent iteration counts (e.g., 47 iterations at h = 2⁻⁸ for Stokes-Navier).
• Theoretical norms derived from the analysis in [24] are successfully extended to include the Lagrange multiplier, enabling robust preconditioning.
• The framework is validated on two benchmark problems: Darcy-Stokes and fluid-structure interaction, with error convergence rates matching theoretical estimates.