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

Characterization of the Solvability of Generalized Constrained Variational Equations

In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.This research is partially supported by the Junta de Andaluca Grant FQM359

A Domain Decomposition Method for the Steady-State Navier-Stokes-Darcy Model with Beavers-Joseph Interface Condition

This paper proposes and analyzes a Robin-type multiphysics domain decomposition method (DDM) for the steady-state Navier-Stokes-Darcy model with three interface conditions. In addition to the two regular interface conditions for the mass conservation and the force balance, the Beavers-Joseph condition is used as the interface condition in the tangential direction. The major mathematical difficulty in adopting the Beavers-Joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system [Y. Cao et al., Comm. Math. Sci., 8 (2010), pp. 1-25; Y. Cao et al., SIAM J. Numer. Anal., 47 (2010), pp. 4239-4256]. In this paper, the well-posedness of the Navier-Stokes-Darcy model with Beavers-Joseph condition is analyzed by using a branch of nonsingular solutions. By following the idea in [Y. Cao et al., Numer. Math., 117 (2011), pp. 601-629], the three physical interface conditions are utilized together to construct the Robin-type boundary conditions on the interface and decouple the two physics which are described by Navier-Stokes and Darcy equations, respectively. Then the corresponding multiphysics DDM is proposed and analyzed. Three numerical experiments using finite elements are presented to illustrate the features of the proposed method and verify the results of the theoretical analysis

A Dual-Porosity-Stokes Model and Finite Element Method for Coupling Dual-Porosity Flow and Free Flow

In this paper, we propose and numerically solve a new model considering confined flow in dual-porosity media coupled with free flow in embedded macrofractures and conduits. Such situation arises, for example, for fluid flows in hydraulic fractured tight/shale oil/gas reservoirs. The flow in dual-porosity media, which consists of both matrix and microfractures, is described by a dual-porosity model. And the flow in the macrofractures and conduits is governed by the Stokes equation. Then the two models are coupled through four physically valid interface conditions on the interface between dual-porosity media and macrofractures/conduits, which play a key role in a physically faithful simulation with high accuracy. All the four interface conditions are constructed based on fundamental properties of the traditional dual-porosity model and the well-known Stokes-Darcy model. The weak formulation is derived for the proposed model, and the well-posedness of the model is analyzed. A finite element semidiscretization in space is presented based on the weak formulation, and four different schemes are then utilized for the full discretization. The convergence of the full discretization with the backward Euler scheme is analyzed. Four numerical experiments are presented to validate the proposed model and demonstrate the features of both the model and the numerical method, such as the optimal convergence rate of the numerical solution, the detail flow characteristics around macrofractures and conduits, and the applicability to the real world problems

Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn—Hilliard—Navier—Stokes equations in the free flow region and Cahn—Hilliard—Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions. © 2018 Society for Industrial and Applied Mathematics

Time-dependent Stokes-Darcy Flow with Deposition

This thesis investigates two nonlinear systems of time-dependent partial differential equations that model a filtration process. Existence and uniqueness results for the governing equations is established. For each system, a finite element scheme capable of approximating the solutions is investigated. Accompanying numerical experiments corroborate the analytical findings. Finally, an optimization application concerning the design of a filter is discussed and supported with a numerical study