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We analyze a parallel, noniterative, multiphysics domain decomposition method for decoupling the Stokes-Darcy model with multistep backward differentiation schemes for the time discretization and finite elements for the spatial discretization. Based on a rigorous analysis of the Ritz projection error shown in this article, we prove almost optimal L² convergence of the numerical solution. In order to estimate the Ritz projection error on the interface, which plays a key role in the error analysis of the Stokes-Darcy problem, we derive L∞ error estimate of the Stokes-Ritz projection under the stress boundary condition for the first time in the literature. The k-step backward differentiation schemes, which are important to improve the accuracy in time discretization with unconditional stability, are analyzed in a general framework for any k ≤ 5. The unconditional stability and high accuracy of these schemes can allow relatively larger time step sizes for given accuracy requirements and hence save a significant amount of computational cost. [ABSTRACT FROM AUTHOR]

In this work we consider the coupled problem of Darcy flow in a fracture and the surrounding porous medium. The fracture is represented as a (d -1)-dimensional interface, and it is nonmatching with the computational grid thanks to a suitable extended finite element method (XFEM) enrichment of the mixed finite element spaces. In the existing literature well posedness has been proven for the discrete problem in the hypothesis of a given solution in the fracture. This work provides theoretical results on the stability and convergence of the discrete, fully coupled problem, yielding sharp conditions on the fracture geometry and on the computational grid to ensure that the inf-sup condition is satisfied by the enriched spaces, as confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

The interface control domain decomposition (ICDD) method of Discacciati, Gervasio, and Quarteroni [SIAM J. Control Optim., 51 (2013), pp. 3434-3458, doi:10.1137/120890764; J. Coupled Syst. Multiscale Dyn., 1 (2013), pp. 372-392, doi:10.1166/jcsmd.2013.1026] is proposed here to solve the coupling between Stokes and Darcy equations. According to this approach, the problem is formulated as an optimal control problem whose control variables are the traces of the velocity and the pressure on the internal boundaries of the subdomains that provide an overlapping decomposition of the original computational domain. A theoretical analysis is carried out, and the well-posedness of the problem is proved under certain assumptions on both the geometry and the model parameters. An efficient solution algorithm is proposed, and several numerical tests are implemented. Our results show the accuracy of the ICDD method, its computational efficiency, and robustness with respect to the different parameters involved (grid size, polynomial degrees, permeability of the porous domain, thickness of the overlapping region). The ICDD approach turns out to be more versatile and easier to implement than the celebrated model based on the Beavers, Joseph, and Saffman coupling conditions. [ABSTRACT FROM AUTHOR]

A two-grid block-centered finite difference method is proposed for solving the two-dimensional Darcy-Forchheimer model describing non-Darcy flow in porous media. To construct the two-grid method we modify the original nonlinear elliptic operator of Darcy-Forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter ε. By using the two-grid method, solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid. Optimal order error estimates for pressure and velocity in discrete L² norms are obtained. Some numerical examples are given to show the accuracy and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

In this paper we devise and analyze a mixed finite element method for a modified Cahn-Hilliard equation coupled with a nonsteady Darcy-Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the discrete phase variable is bounded in L∞ (0, T;L∞) and the discrete chemical potential is bounded in L∞(0, T; L²), for any time and space step sizes, in two and three dimensions, and for any finite final time T. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a single-component model of heat and vapor (sweat) transport through three-dimensional porous textile materials with phase change, which is described by a nonlinear, degenerate, and strongly coupled parabolic system. An uncoupled (splitting) Galerkin method with semi-implicit Euler scheme in time direction is proposed for the system. In this method, a linearized scheme is applied for the approximation to Darcy's velocity simultaneously in the mass and energy equations, which leads to physical conservation of the method in the flow convection. The existence and uniqueness of solution of the finite element system is proved and the optimal error estimate in an energy norm is obtained. Numerical results are presented to confirm our theoretical analysis and are compared with experimental data. [ABSTRACT FROM AUTHOR]

We introduce a new class of higher order mixed finite volume methods for elliptic problems. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, u = -K ∇ p. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed finite volume methods of higher order. We show that these new schemes are equivalent to the nonconforming finite element spaces used to define them. The Darcy velocity can be locally recovered from the solution of nonconforming finite element method. Hence our work opens a way to make higher order mixed method more practicable. Three-dimensional extensions to parallelepiped elements are also presented. This work can be viewed as a generalization of earlier works which were devoted to reducing the (lowest order) mixed finite element method to a corresponding nonconforming finite element method. An optimal error analysis is carried out and numerical results are presented which confirm the theory. [ABSTRACT FROM AUTHOR]

We study numerical methods for a coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. A decoupled and linearized two-grid algorithm is proposed. Numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm. [ABSTRACT FROM AUTHOR]

We consider a stochastic Darcy's pressure equation whose coefficient is generated by a white noise process on a Hilbert space employing the ordinary (rather than the Wick) product. A weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf-sup condition and well-posedness of the problem. We generalize the numerical approximations proposed in Benth and Theting [Stochastic Anal. Appl., 20 (2002), pp. 1191-1223] for Wick stochastic partial differential equations to the ordinary product stochastic pressure equation. We establish discrete inf-sup conditions and provide a priori error estimates for a wide class of norms. The proposed numerical approximation is based on Wiener-Chaos finite element methods and yields a positive definite symmetric linear system. We also improve and generalize the approximation results of Benth and Gjerde [Stochastics Stochastics Rep., 63 (1998), pp. 313-326] and Cao [Stochastics, 78 (2006), pp. 179-187] when a (generalized) process is truncated by a finite Wiener-Chaos expansion. Finally, we present numerical experiments to validate the results. [ABSTRACT FROM AUTHOR]

This work introduces and analyzes novel stable Petrov-Galerkin enriched methods (PGEM) for the Darcy problem, based on the simplest but unstable continuous P1/P0 pair. Stability is recovered inside a Petrov-Galerkin framework where elementwise dependent residual functions, named multiscale functions, enrich both velocity and pressure trial spaces. Unlike the velocity test space that is augmented with bubble-like functions, multiscale functions correct edge residuals as well. The multiscale functions turn out to be the well-known lowest order Raviart-Thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure. The enrichment strategy suggests the way to recover the local mass conservation property for nodal-based interpolation spaces. We prove that the method and its symmetric version are well-posed and achieve optimal error estimates in natural norms. Numerical validations confirm claimed theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart from the interest of this fact, the important issue is that we are able to deal with both problems at the same time, in a completely unified manner, in spite of the fact that the functional setting is different. Concerning the stabilization formulation, we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters, both in what refers to stability and to accuracy. This choice is shown to be crucial in the case of Darcy's problem. As an additional feature of this work, we treat two types of stabilized formulations, showing that they have a very similar behavior. [ABSTRACT FROM AUTHOR]

In this work, we couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additional smallness restrictions. Then we propose a discontinuous Galerkin scheme for discretizing the equations and do its numerical analysis. [ABSTRACT FROM AUTHOR]

We study numerical methods for solving a coupled Stokes-Darcy problem in porous media flow applications. A two-grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions. Error estimates are derived for the proposed method. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid approach for solving multimodeling problems. Potential extensions and future directions are discussed. [ABSTRACT FROM AUTHOR]

In this paper we consider a coupled system made of the Stokes and Darcy equations, and we propose some iteration-by-subdomain methods based on Robin conditions on the interface. We prove the convergence of these algorithms, and for suitable finite element approximations we show that the rate of convergence is independent of the mesh size h. Special attention is paid to the optimization of the performance of the methods when both the kinematic viscosity ? of the fluid and the hydraulic conductivity tensor K of the porous medium are very small. [ABSTRACT FROM AUTHOR]

We consider control-volume mixed finite element methods for the approximation of second-order elliptic problems on rectangular grids. These methods associate control volumes (covolumes) with the vector variable as well as the scalar, obtaining local algebraic representation of the vector equation (e.g., Darcy's law) as well as the scalar equation (e.g., conservation of mass). We establish O(h²) superconvergence for both the scalar variable in a discrete L²-norm and the vector variable in a discrete H(div)-norm. The analysis exploits a relationship between control-volume mixed finite element methods and the lowest order Raviart--Thomas mixed finite element methods. [ABSTRACT FROM AUTHOR]

A locally conservative numerical method for solving the coupled Stokes and Darcy flows problem is formulated and analyzed. The approach employs the mixed finite element method for the Darcy region and the discontinuous Galerkin method for the Stokes region. A discrete inf-sup condition and optimal error estimates are derived. [ABSTRACT FROM AUTHOR]

In this paper, we prove the convergence of a numerical method for solving two-phase immiscible, incompressible flow in porous media. The method combines an upwind time implicit finite volume scheme for the saturation equation (hyperbolic-parabolic type) and a centered finite volume scheme for the Chavent global pressure equation (elliptic type). The capillary pressure is not neglected, and we study the case when the diffusion term in the saturation equation is weakly degenerated. Estimates on the approximate solution are proven; then by using compactness theorems we obtain a limit when the size of the discretization goes to zero, and we prove that this limit is the unique weak solution of the problem that we study. [ABSTRACT FROM AUTHOR]