A Vanka-based parameter-robust multigrid relaxation for the Stokes–Darcy Brinkman problems

We consider a block-structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element-based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka-based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two-grid and V(1,1)-cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter.     

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.     

Decoupled modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem

In this paper, a modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem with the Beavers–Joseph–Saffman interface condition is presented. In this method, the Navier–Stokes/Darcy equation is decoupled into two equations, one is the Navier–Stokes equation, the other is the Darcy equation, and the Navier–Stokes equation is solved by the modified characteristics finite element method. The theory analysis shows that this method has a good convergence property. In order to show the effect of our method, some numerical results was presented. The numerical results show that this method is highly efficient. Copyright © 2013 John Wiley & Sons, Ltd.     

Spectral discretization of Darcy equations coupled with Stokes equations by vorticity–velocity–pressure formulations

We propose to make the numerical analysis of a model coupling the Darcy equations in a porous medium with the Stokes equations in the cracks. The coupling is provided by a pressure continuity on the interface. We describe a discretization by spectral element methods. We derive a priori optimal error estimates and we present some numerical experiments which confirm the results of the analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1628–1651, 2017     

On the well-posed coupling between free fluid and porous viscous flows

We present a well-posed model for the Stokes/Brinkman problem with a family of jump embedded boundary conditions (J.E.B.C.) on an immersed interface with weak regularity assumptions. It arises from a general framework recently proposed for fictitious domain problems. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface $\Sigma$ separating the fluid and porous domains. These conditions are well chosen to get the coercivity of the operator. Then, the general framework allows us to prove new results on the global solvability of some models with physically relevant stress or velocity jump boundary conditions for the momentum transport at a fluid–porous interface. The Stokes/Brinkman problem with Ochoa-Tapia and Whitaker (1995) [9], [10] interface conditions and the Stokes/Darcy problem with Beavers and Joseph (1967) [13] conditions are both proved to be well-posed, by an asymptotic analysis. Up to now, only the Stokes/Darcy problem with Saffman (1971) [15] approximate interface conditions with negligible tangential porous velocity was known to be well-posed.     

Convergence of a family of Galerkin discretizations for the Stokes–Darcy coupled problem

In this article we analyze the well‐posedness (unique solvability, stability, and Céa's estimate) of a family of Galerkin schemes for the coupling of fluid flow with porous media flow. Flows are governed by the 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 usual primal formulation in the Stokes domain and the dual‐mixed one in the Darcy region, which yields a compact perturbation of an invertible mapping as the resulting operator equation. We then apply a classical result on projection methods for Fredholm operators of index zero to show that use of any pair of stable Stokes and Darcy elements implies the well‐posedness of the corresponding Stokes—Darcy Galerkin scheme. This extends previous results showing well‐posedness only for Bernardi—Raugel and Raviart—Thomas elements. In addition, we show that under somewhat more demanding hypotheses, an alternative approach that makes no use of compactness arguments can also be applied. Finally, we provide several numerical results illustrating the good performance of the Galerkin method for different geometries of the problem using the MINI element and the Raviart—Thomas subspace of lowest order. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 721–748, 2011     

Numerical analysis for the mixed Navier–Stokes and Darcy Problem with the Beavers–Joseph interface condition

In this article, we consider the coupled Navier–Stokes and Darcy problem with the Beavers–Joseph interface condition. With suitable restrictions of physical parameters α and ν, we prove the existence and local uniqueness of a weak solution. Then we propose a coupled finite element scheme and a decoupled and linearized scheme based on two‐grid finite element. Under suitable further restrictions, their optimal error estimates are obtained. Finally numerical experiments indicate the validity of the theoretical results as well as the efficiency and effectiveness of the decoupled and linearized two‐grid algorithm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1009–1030, 2015     

Partitioned penalty methods for the transport equation in the evolutionary Stokes–Darcy‐transport problem

There has been a surge of work on models for coupling surface‐water with groundwater flows which is at its core the Stokes–Darcy problem, as well as methods for uncoupling the problem into subdomain, subphysics solves. The resulting (Stokes–Darcy) fluid velocity is important because the flow transports contaminants. The numerical analysis and algorithm development for the evolutionary transport problem has, however, focused on a quasi‐static Stokes–Darcy model and a single domain (fully coupled) formulation of the transport equation. This report presents a numerical analysis of a partitioned method for contaminant transport for the fully evolutionary system. The algorithm studied is unconditionally stable with one subdomain solve per step. Numerical experiments are given using the proposed algorithm that investigates the effects of the penalty parameters on the convergence of the approximations.     

Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in Lipschitz domains

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in , with small boundary datum in L²‐based Sobolev spaces. A useful intermediary result is the well‐posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in , with Dirichlet boundary condition and data in L²‐based Sobolev spaces. We obtain this well‐posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well‐posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.     

Integral representation of a solution to the Stokes–Darcy problem

With methods of potential theory, we develop a representation of a solution of the coupled Stokes–Darcy model in a Lipschitz domain for boundary data in H^?1/2. Copyright © 2014 John Wiley & Sons, Ltd.     

A posteriori error estimate for the Stokes–Darcy system

We consider the a posteriori error estimates for finite element approximations of the Stokes–Darcy system. The finite element spaces adopted are the Hood–Taylor element for the velocity and the pressure in fluid region and conforming piecewise quadratic element for the pressure in porous media region. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. Copyright © 2011 John Wiley & Sons, Ltd.     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Integral potential method for a transmission problem with Lipschitz interface in \mathbb{R}^{3} for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the nonlinear Darcy–Forchheimer–Brinkman system and the linear Stokes system in two complementary Lipschitz domains in \({\mathbb{R}^{3}}\), one of them is a bounded Lipschitz domain \({\Omega}\) with connected boundary, and the other one is the exterior Lipschitz domain \({\mathbb{R}^{3} \setminus \overline{\Omega }}\). We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces.     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

Superconvergence analysis of FEMs for the Stokes–Darcy system

We consider a superconvergence analysis for quadratic finite element approximations of the Stokes–Darcy system. The superclose property of an extra half order is proven for uniform triangular meshes. Based on the result of the superclose property, global superconvergence is derived by applying a postprocessing technique. In addition, some numerical examples are presented to demonstrate our theoretical results. Copyright © 2010 John Wiley & Sons, Ltd.     

A decoupling two‐grid algorithm for the mixed Stokes‐Darcy model with the Beavers‐Joseph interface condition

In this article, a decoupling scheme based on two‐grid finite element for the mixed Stokes‐Darcy problem with the Beavers‐Joseph interface condition is proposed and investigated. With a restriction of a physical parameter α, we derive the numerical stability and error estimates for the scheme. Numerical experiments indicate that such two‐grid based decoupling finite element schemes are feasible and efficient. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1066–1082, 2014     

Optimal estimates on stabilized finite volume methods for the incompressible Navier–Stokes model in three dimensions

Optimal estimates on stabilized finite volume methods for the three dimensional Navier–Stokes model are investigated and developed in this paper. Based on the global existence theorem [23], we first prove the global bound for the velocity in the H¹‐norm in time of a solution for suitably small data, and uniqueness of a suitably small solution by contradiction. Then, a full set of estimates is then obtained by some classical Galerkin techniques based on the relationship between finite element methods and finite volume methods approximated by the lower order finite elements for the three dimensional Navier–Stokes model.     

Kronecker product approximation preconditioners for convection–diffusion model problems

We consider the iterative solution of linear systems arising from four convection–diffusion model problems: scalar convection–diffusion problem, Stokes problem, Oseen problem and Navier–Stokes problem. We design preconditioners for these model problems that are based on Kronecker product approximations (KPAs). For this we first identify explicit Kronecker product structure of the coefficient matrices, in particular for the convection term. For the latter three model cases, the coefficient matrices have a 2 × 2 block structure, where each block is a Kronecker product or a summation of several Kronecker products. We then use this structure to design a block diagonal preconditioner, a block triangular preconditioner and a constraint preconditioner. Numerical experiments show the efficiency of the three KPA preconditioners, and in particular of the constraint preconditioner that usually outperforms the other two. This can be explained by the relationship that exists between these three preconditioners: the constraint preconditioner can be regarded as a modification of the block triangular preconditioner, which at its turn is a modification of the block diagonal preconditioner based on the cell Reynolds number. Copyright © 2009 John Wiley & Sons, Ltd.     

On the linear stability analysis of a Maxwell fluid with double-diffusive convection

The problem of double-diffusive convection and cross-diffusion in a Maxwell fluid in a horizontal layer in porous media is re-examined using the modified Darcy–Brinkman model. The effect of Dufour and Soret parameters on the critical Darcy–Rayleigh numbers is investigated. Analytical expressions of the critical Darcy–Rayleigh numbers for the onset of stationary and oscillatory convection are derived. Numerical simulations show that the presence of Dufour and Soret parameters has a significant effect on the critical Darcy–Rayleigh number for over-stability. In the limiting case some previously published results are recovered.     

Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity, and pressure with non-constant viscosity. The analysis is performed by the classical Babu?ka–Brezzi theory, and we state that any inf–sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates, which are further confirmed through computational examples.