Augmented interface systems for the Darcy-Stokes problem

In this paper we study interface equations associated to the Darcy-Stokes problem using the classical Steklov-Poincaré approach and a new one called augmented. We compare these two families of methods and characterize at the discrete level suitable preconditioners with additive and multiplicative structures. Finally, we present some numerical results to assess their behavior in presence of small physical parameters

Numerical approximation of a steady MHD problem

We consider a magnetohydrodynamic (MHD) problem which models the steady flow of a conductive incompressible fluid confined in a bounded region and subject to the Lorentz force exerted by the interaction of electric currents and magnetic fields. We present an iterative method inspired to operator splitting to solve this nonlinear coupled problem, and a discretization based on conforming finite elements

Navier-Stokes/Forchheimer models for filtration through porous media

Modeling the filtration of incompressible fluids through porous media requires dealing with different types of partial differential equations in the fluid and porous subregions of the computational domain. Such equations must be coupled through physically significant continuity conditions at the interface separating the two subdomains. To avoid the difficulties of this heterogeneous approach, a widely used strategy is to consider the Navier–Stokes equations in the whole domain and to correct them introducing suitable terms that mimic the presence of the porous medium. In this paper we discuss these two different methodologies and we compare them numerically on a sample test case after proposing an iterative algorithm to solve a Navier–Stokes/Forchheimer problem. Finally, we apply these strategies to a problem of internal ventilation of motorbike helmets

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier–Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the standard mixed formulation in the Navier–Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported

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

Optimized Schwarz methods for the Stokes-Darcy coupling

This paper 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

The interface control domain decomposition (ICDD) method for elliptic problems

Interface controls are unknown functions used as Dirichlet or Robin boundary data on the interfaces of an overlapping decomposition designed for solving second order elliptic boundary value problems. The controls are computed through an optimal control problem with either distributed or interface observation. Numerical results show that, when interface observation is considered, the resulting interface control domain decomposition method is robust with respect to coefficients variations; it can exploit nonconforming meshes and provides optimal convergence with respect to the discretization parameters; finally it can be easily used to face heterogeneous advection--advection-diffusion couplings

Computational modeling of coupled free and porous media flow for membrane-based filtration systems: a review

We review different mathematical models proposed in literature to describe fluid-dynamic aspects in membrane-based water filtration systems. Firstly, we discuss the societal impact of water filtration, especially in the context of developing countries under emergency situations, and then review the basic concepts of membrane science that are necessary for a mathematical description of a filtration system. Secondly, we categorize the mathematical models available in the literature as (a) microscopic, if the pore-scale geometry of the membrane is accounted for; (b) reduced, if the membrane is treated as a geometrically lower-dimensional entity due to its small thickness compared to the free flow domain; (c) mesoscopic, if the characteristic geometrical dimension of the free flow domain and the porous domain is the same, and a multi-physics problem involving both incompressible fluid flow and porous media flow is considered. Implementation aspects of mesoscopic models in CFD software are also discussed with the help of relevant examples

Domain decomposition methods for domain composition purpose: Chimera, overset, gluing and sliding mesh methods

Domain composition methods (DCM) consist in obtaining a solution to a problem, from the formulations of the same problem expressed on various subdomains. These methods have therefore the opposite objective of domain decomposition methods (DDM). Indeed, in contrast to DCM, these last techniques are usually applied to matching meshes as their purpose consists mainly in distributing the work in parallel environments. However, they are sometimes based on the same methodology as after decomposing, DDM have to recompose. As a consequence, in the literature, the term DDM has many times substituted DCM. DCM are powerful techniques that can be used for different purposes: to simplify the meshing of a complex geometry by decomposing it into different meshable pieces; to perform local refinement to adapt to local mesh requirements; to treat subdomains in relative motion (Chimera, sliding mesh); to solve multiphysics or multiscale problems, etc. The term DCM is generic and does not give any clue about how the fragmented solutions on the different subdomains are composed into a global one. In the literature, many methodologies have been proposed: they are mesh-based, equation-based, or algebraic-based. In mesh-based formulations, the coupling is achieved at the mesh level, before the governing equations are assembled into an algebraic system (mesh conforming, Shear-Slip Mesh Update, HERMESH). The equation-based counterpart recomposes the solution from the strong or weak formulation itself, and are implemented during the assembly of the algebraic system on the subdomain meshes. The different coupling techniques can be formulated for the strong formulation at the continuous level, for the weak formulation either at the continuous or at the discrete level (iteration-by-subdomains, mortar element, mesh free interpolation). Although the different methods usually lead to the same solutions at the continuous level, which usually coincide with the solution of the problem on the original domain, they have very different behaviors at the discrete level and can be implemented in many different ways. Eventually, algebraic- based formulations treat the composition of the solutions directly on the matrix and right-hand side of the individual subdomain algebraic systems. The present work introduces mesh-based, equation-based and algebraicbased DCM. It however focusses on algebraic-based domain composition methods, which have many advantages with respect to the others: they are relatively problem independent; their implicit implementation can be hidden in the iterative solver operations, which enables one to avoid intensive code rewriting; they can be implemented in a multi-code environment

Iterative methods for Stokes/Darcy coupling

We present iterative subdomain methods based on a domain decomposition approach to solve the coupled Stokes/Darcy problem using finite elements. The dependence of the convergence rate on the grid parameter h and on the physical data is discussed; some difficulties encountered when applying the algorithms are indicated together with possible improvement strategies