Is minimizing the convergence rate a good choice for efficient Optimized Schwarz preconditioning in heterogeneous coupling? The Stokes-Darcy case

Optimized Schwarz Methods (OSM) are domain decomposition techniques based on Robin-type interface condition that have became increasingly popular in the last two decades. Ensuring convergence also on non-overlapping decompositions, OSM are naturally advocated for the heterogeneous coupling of multiphysics problems. Classical approaches optimize the coefficients in the Robin condition by minimizing the effective convergence rate of the resulting iterative algorithm. However, when OSM are used as preconditioners for Krylov solvers of the resulting interface problem, such parameter optimization does not necessarily guarantee the fastest convergence. This drawback is already known for homogeneous decomposition, but in the case of heterogeneous decomposition, the poor performance of the classical optimization approach becomes utterly evident. In this paper, we highlight this drawback for the Stokes/Darcy problem and we propose a more effective alternative optimization procedure

Is minimising the convergence rate the best choice for efficient Optimized Schwarz preconditioning in heterogeneous coupling? The Stokes-Darcy case

Optimized Schwarz Methods (OSM) are domain decomposition techniques based on Robin-type interface condition that have became increasingly popular in the last two decades. Ensuring convergence also on non-overlapping decompositions, OSM are naturally advocated for the heterogeneous coupling of multi-physics problems. Classical approaches optimize the coefficients in the Robin condition by minimizing the effective convergence rate of the resulting iterative algorithm. However, when OSM are used as preconditioners for Krylov solvers of the resulting interface problem, such parameter optimization does not necessarily guarantee the fastest convergence. This drawback is already known for homogeneous decomposition, but in the case of heterogeneous decomposition, the poor performance of the classical optimization approach becomes utterly evident. In this paper, we highlight this drawback for the Stokes/Darcy problem and we propose a more effective alternative optimization procedure.European Union Seventh Framework Programme (FP7/2007-2013; grant 294229) to M. Discacciat

Domain decomposition methods for the coupling of surface and groundwater flows

The purpose of this thesis is to investigate, from both the mathematical and numerical viewpoint, the coupling of surface and porous media flows, with particular concern on environmental applications. Domain decomposition methods are applied to set up effective iterative algorithms for the numerical solution of the global problem. To this aim, we reformulate the coupled problem in terms of an interface (Steklov-Poincaré) equation and we investigate the properties of the Steklov-Poincaré operators in order to characterize optimal preconditioners that, at the discrete level, yield convergence in a number of iterations independent of the mesh size h. We consider a new approach to the classical Robin-Robin method and we reinterpret it as an alternating direction iterative algorithm. This allows us to characterize robust preconditioners for the linear Stokes/Darcy problem which improve the behaviour of the classical Dirichlet- Neumann and Neumann-Neumann ones. Several numerical tests are presented to assess the convergence properties of the proposed algorithms. Finally, the nonlinear Navier-Stokes/Darcy coupling is investigated and a general nonlinear domain decomposition strategy is proposed for the solution of the interface problem, extending the usual Newton or fixed-point based algorithms

A Parallel Robin-Robin Domain Decomposition Method for the Stokes-Darcy System

We propose a new parallel Robin-Robin domain decomposition method for the coupled Stokes-Darcy system with Beavers-Joseph-Saffman-Jones interface boundary condition. in particular, we prove that, with an appropriate choice of parameters, the scheme converges geometrically independent of the mesh size. © 2011 Society for Industrial and Applied Mathematics

Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method

Optimized Schwarz Methods in the Stokes-Darcy Coupling

This article studies optimized Schwarz methods for the Stokes–Darcy problem. Robin transmission conditions are introduced, and the coupled problem is reduced to a suitable interface system that can be solved using Krylov methods. Practical strategies to compute optimal Robin coefficients are proposed, which take into account both the physical parameters of the problem and the mesh size. Numerical results show the effectiveness of our approach.European Union Seventh Framework Programme (FP7/2007-2013; grant 294229) to M. Discacciat

On iterative subdomain methods for the Stokes--Darcy problem

Iterative subdomain methods for the Stokes--Darcy problem that use Robin boundary conditions on the interface are reviewed. Their common underlying structure and their main differences are identified. In particular, it is clarified that there are different updating strategies for the interface conditions. For small values of fluid viscosity and hydraulic permeability, which are relevant in applications from geosciences, it is shown in numerical studies that only one of these updating strategies leads to an efficient numerical method, if this strategy is used in combination with appropriate parameters in the Robin boundary conditions. In particular, it is observed that the values of appropriate parameters are larger than those proposed so far. Not only the size but also the ratio of appropriate Robin parameters depends on the coefficients of the problem

Modeling and a Domain Decomposition Method with Finite Element Discretization for Coupled Dual-Porosity Flow and Navier–Stokes Flow

In This Paper, We First Propose and Analyze a Steady State Dual-Porosity-Navier–Stokes Model, Which Describes Both Dual-Porosity Flow and Free Flow (Governed by Navier–Stokes Equation) Coupled through Four Interface Conditions, Including the Beavers–Joseph Interface Condition. Then We Propose a Domain Decomposition Method for Efficiently Solving Such a Large Complex System. Robin Boundary Conditions Are Used to Decouple the Dual-Porosity Equations from the Navier–Stokes Equations in the Coupled System. based on the Two Decoupled Sub-Problems, a Parallel Robin-Robin Domain Decomposition Method is Constructed and Then Discretized by Finite Elements. We Analyze the Convergence of the Domain Decomposition Method with the Finite Element Discretization and Investigate the Effect of Robin Parameters on the Convergence, Which Also Provide Instructions for How to Choose the Robin Parameters in Practice. Three Cases of Robin Parameters Are Studied, Including a Difficult Case Which Was Not Fully Addressed in the Literature, and the Optimal Geometric Convergence Rate is Obtained. Numerical Experiments Are Presented to Verify the Theoretical Conclusions, Illustrate How the Theory Can Provide Instructions on Choosing Robin Parameters, and Show the Features of the Proposed Model and Domain Decomposition Method