Stokes Darcy Problem Research Articles

A unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem

A unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem

A penalty method for approximation of the stationary Stokes–Darcy problem

In this work, the penalty method is studied for the mixed Stokes–Darcy problem, motivated by the penalty method applied to Stokes equation. This work first proposes the penalty Stokes–Darcy model at the continuous level. Then we prove that the solution of the penalty model converges strongly to the original solution as Oϵ in which the penalty parameter is ϵ→0. What is more, the finite element method is used to solve the penalty model and the optimal error estimates are presented. Finally, several numerical tests are carried out to verify our theoretical results.

Optimal long time error estimates of a second-order decoupled finite element method for the Stokes–Darcy problem

In this paper, we propose a second-order decoupled finite element method based on lagging a part of the interfacial coupling terms for the time dependent Stokes–Darcy problem, which only need to solve two sub-physical problems sequentially. Under a modest time step restriction Δt≤C(physical parameters), the optimal long time error estimates are obtained both in the L2 norm and in the H1 norm. Numerical results are provided to illustrate the convergence rate O(Δt2) of the temporal approximation.

A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

Optimized Schwarz methods for the time-dependent Stokes–Darcy coupling

Abstract This paper derives optimized coefficients for optimized Schwarz iterations for the time-dependent Stokes–Darcy problem using an innovative strategy to solve a nonstandard min-max problem. The coefficients take into account both physical and discretization parameters that characterize the coupled problem, and they guarantee the robustness of the associated domain decomposition method. Numerical results validate the proposed approach in several test cases with physically relevant parameters.

Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes–Darcy flow problems

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy’s law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations.

A pressure-robust mixed finite element method for the coupled Stokes–Darcy problem

A pressure-robust numerical method for the coupled Stokes–Darcy problem is proposed. We use the continuous piecewise linear polynomial space combined with the lowest order Raviart–Thomas space to discretize the Stokes equation in the fluid region and the lowest order Raviart–Thomas elements to discretize the Darcy equation in the porous media domain. We also perform a divergence-free reconstruction of the test function for the right-hand term. Our method has good properties of convergence, pressure-robustness and local mass conservation in the sense of projection. A discrete inf-sup condition and error estimates for the numerical scheme are demonstrated, the error of velocity is independent of viscosity and pressure. Finally, the theoretical analysis is validated with several examples, demonstrating the superiority of the pressure-robust scheme.

On the Conditions at the Interface Between Homogeneous and Porous Environments for the Stokes–Darcy Problem

On the Conditions at the Interface Between Homogeneous and Porous Environments for the Stokes–Darcy Problem

Efficient coupled deep neural networks for the time-dependent coupled Stokes-Darcy problems

In this paper, we propose and investigate an efficient method called CDNNs (Coupled Deep Neural Networks) for the time-dependent coupled Stokes-Darcy problems. Specifically, we encode complex interface conditions related to the variables of the coupled problems into several neural networks to constrain the approximation solution. We define a custom loss function to guarantee the physical properties of the numerical solution as well as the conservation of the energy. In particular, the present method is mesh-free since it only inputs random spatiotemporal points and can avoid the difficulties and complexities caused by the mesh-based method. Moreover, our method is parallel, it solves each variable simultaneously and independently. Furthermore, we obtain the convergence analysis to illustrate the capabilities of our method for solving the coupled problems. Numerical experiments further demonstrate the accuracy and efficiency of the proposed method.

A Robin-Type Domain Decomposition Method for a Novel Mixed-Type DG Method for the Coupled Stokes--Darcy Problem

In this paper, we first propose and analyze a novel mixed-type DG method for the coupled Stokes-Darcy problem on simplicial meshes. The proposed formulation is locally conservative. A mixed-type DG method in conjunction with the stress-velocity formulation is employed for the Stokes equations, where the symmetry of stress is strongly imposed. The staggered DG method is exploited to discretize the Darcy equations. As such, the discrete formulation can be easily adapted to account for the Beavers-Joseph-Saffman interface conditions without introducing additional variables. Importantly, the continuity of normal velocity is satisfied exactly at the discrete level. A rigorous convergence analysis is performed for all the variables. Then we devise and analyze a domain decomposition method via the use of Robin-type interface boundary conditions, which allows us to solve the Stokes subproblem and the Darcy subproblem sequentially with low computational costs. The convergence of the proposed iterative method is analyzed rigorously. In particular, the proposed iterative method also works for very small viscosity coefficient. Finally, several numerical experiments are carried out to demonstrate the capabilities and accuracy of the novel mixed-type scheme, and the convergence of the domain decomposition method.

Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form

We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and non-conforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.

Analysis of the Stokes–Darcy problem with generalised interface conditions

Fluid flows in coupled systems consisting of a free-flow region and the adjacent porous medium appear in a variety of environmental settings and industrial applications. In many applications, fluid flow is non-parallel to the fluid–porous interface that requires a generalisation of the Beavers–Joseph coupling condition typically used for the Stokes–Darcy problem. Generalised coupling conditions valid for arbitrary flow directions to the interface are recently derived using the theory of homogenisation and boundary layers. The aim of this work is the mathematical analysis of the Stokes–Darcy problem with these generalised interface conditions. We prove the existence and uniqueness of the weak solution of the coupled problem. The well-posedness is guaranteed under a suitable relationship between the permeability and the boundary layer constants containing geometrical information about the porous medium and the interface. We study the validity of the obtained results for realistic problems numerically and provide a benchmark for numerical solution of the Stokes–Darcy problem with generalised interface conditions.

Unconditional stability over long time intervals of a two-level coupled MacCormack/Crank–Nicolson method for evolutionary mixed Stokes-Darcy model

This paper analyzes the stability of a two-level coupled explicit MacCormack/Crank–Nicolson method for solving the nonstationary mixed Stokes-Darcy model. The approach is systematically derived based on the monolithic weak formulations of the explicit MacCormack technique and the implicit Crank–Nicolson discretization. The stability of the explicit scheme does not require a time step restriction. The two-step MacCormack provides an approximate solution at the coarse grid level while the implicit Crank–Nicolson algorithm uses this approximation to compute the desired numerical solution at the fine grid stage. The proposed method is unconditionally stable over long time intervals and the computational cost is reduced. This combination is efficient than a wide set of numerical schemes applied to time dependent mixed Stokes-Darcy problem. A large range of numerical examples which confirm the theoretical study are presented to illustrate the performance of the proposed method.

A second order multirate scheme for the evolutionary Stokes–Darcy model

A second order multirate scheme for the evolutionary coupled Stokes–Darcy problem is proposed and analyzed. The scheme decouples the problem into two smaller subproblems at each time step by exploiting the mismatch in flow activity between the domains. Specifically, the slower Darcy subproblem is integrated with a larger time step compared to the more active Stokes subproblem. This avoids expending unnecessary computational effort on sampling the slow changing Darcy solution. The slow varying Darcy solution is extrapolated/interpolated onto the fine grid for the integration of the Stokes subproblem. The long-time stability and convergence of the method is proved under a time step restriction that depends on physical parameters and the ratio of the time steps in each subdomain. The scheme includes a stabilization term that relaxes the time step restriction. Numerical experiments confirm convergence of the method and the effectiveness of the stabilization technique at relaxing the stability time step restriction.

Modified formulation of the interfacial boundary condition for the coupled Stokes–Darcy problem

Modified formulation of the interfacial boundary condition for the coupled Stokes–Darcy problem

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation. A discrete inf-sup condition is proved and the optimal error estimates are also derived. Numerical experiments validate the theoretical analysis.

A novel local and parallel finite element method for the mixed Navier–Stokes–Darcy problem

In this paper, based on two-grid discretization, a new local and parallel finite element method is proposed and investigated for the mixed Navier–Stokes–Darcy problem. Compared with the classical local and parallel finite element method, the main features of our method are: (1) partition of unity method is considered to generate local subproblems; (2) global continuous approximations are achieved. We theoretically analyze the proposed method and report on some numerical tests to support the theoretical results.

A New Unified Stabilized Mixed Finite Element Method of the Stokes-Darcy Coupled Problem: Isotropic Discretization

In this paper, we develop an a-priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in RN, N ∈ {2,3}, on isotropic meshes. 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. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.

Robust preconditioning for coupled Stokes–Darcy problems with the Darcy problem in primal form

The coupled Darcy–Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolithic solution algorithms of the coupled Darcy–Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters, i.e. the permeability, viscosity and Beavers–Joseph–Saffman condition. Our approach is similar to that of Holter et al. (2020), but since the Darcy problem is in primal form, expressing mass conservation at the interface involves the normal derivative, which introduces some mathematical challenges. These challenges will be specifically addressed in this paper, in particular we will employ fractional Laplacians at the interface. Numerical experiments illustrating the performance are provided. The preconditioner is posed in non-standard Sobolev spaces which may be perceived as an obstacle for its use in applications. However, we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice.

Nitsche’s type stabilized finite element method for the fully mixed Stokes–Darcy problem with Beavers–Joseph conditions

In this paper, we develop a Nitsche’s type interface stabilized method for fully mixed Stokes–Darcy problem with original Beavers–Joseph conditions. The method introduces two strongly consistent interface stabilization terms to guarantee the stability of the fully mixed finite element method. The stability and optimal error estimates of this new stabilized method are also derived.