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This article presents a mass-conserved Allen-Cahn type two-phase ferrofluid flow model and establishes its corresponding energy law. The model is a highly coupled, nonlinear saddle point system consisting of the mass-conserved Allen-Cahn equation, the Navier-Stokes equation, the magnetostatic equation, and the magnetization equation. We develop a unified framework of the scalar auxiliary variable (SAV) method and the zero energy contribution (ZEC) approach, which constructs a mass-conserved, fully decoupled, second-order accurate in time, and unconditionally energy-stable linear scheme. We incorporate several distinct numerical techniques, including refor- mulations of the equations to remove the linear couplings and implicit nonlocal integration, the projection method to decouple the velocity and pressure, a symmetric implicit-explicit format for symmetric positive definite nonlinearity, and the continuous finite element method discretization. We also analyze the mass-conserved property, unconditional energy stability, and well-posedness of the scheme. To demonstrate the effectiveness, stability, and accuracy of the developed model and numerical algorithm, we implemented several numerical examples, involving a ferrofluid hedgehog in 2D and a ferromagnetic droplet in 3D. It is worth mentioning that the proposed unified framework of the SAV-ZEC method is also applicable to designing efficient schemes for other coupled-type fluid flow phase-field systems. [ABSTRACT FROM AUTHOR]

In this paper we develop and analyze a variational data assimilation method with efficient decoupled iterative numerical algorithms for the Stokes–Darcy equations with the Beavers–Joseph interface condition. By using Tikhonov regularization and formulating the variational data assimilation into an optimization problem, we establish the existence, uniqueness, and stability of the optimal solution. Based on the weak formulation of the Stokes–Darcy equations, the Lagrange multiplier rule is utilized to derive the first order optimality system for both the continuous and discrete variational data assimilation problems, where the discrete data assimilation is based on a finite element discretization in space and the backward Euler scheme in time. By rescaling the optimality system and then analyzing its corresponding bilinear forms, we prove the optimal finite element convergence rate with special attention paid to recovering uncertainties missed in the optimality system. To solve the discrete optimality system efficiently, three decoupled iterative algorithms are proposed to address the computational cost for both well-conditioned and ill-conditioned variational data assimilation problems, respectively. Finally, numerical results are provided to validate the proposed methods. [ABSTRACT FROM AUTHOR]

The two-phase ferrohydrodynamics model consisting of Cahn--Hilliard equations, Navier--Stokes equations, magnetization equations, and magnetostatic equations is a highly nonlinear, coupled, and saddle point structural multiphysics PDE system. While various works exist to develop fully decoupled, linear, second-order in time, and unconditionally energy stable methods for simpler gradient flow models, existing ideas may not be applicable to this complex model or may be only applicable to part of this model. Therefore, significant challenges remain in developing corresponding efficient fully discrete numerical algorithms with the four above-mentioned desired properties, which will be addressed in this paper by dynamically incorporating several key ideas, including a reformulated weak formulation with special test functions for overcoming two major difficulties caused by the magnetostatic equation, the decoupling technique based on the "zeroenergy-contribution"" property to handle the coupled nonlinear terms, the second-order projection method for the Navier--Stokes equations, and the invariant energy quadratization (IEQ) method for the time marching. Among all these ideas, the reformulated weak formulation serves as a key bridge between the existing techniques and the challenges of the target model, with all of the four desired properties kept in mind. We demonstrate the well-posedness of the proposed scheme and rigorously show that the scheme is unconditionally energy stable. Extensive numerical simulations, including accuracy/stability tests, and several 2D/3D benchmark Rosensweig instability problems for "spiking" phenomena of ferrofluids are performed to verify the effectiveness of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the design of optimized Schwarz domain decomposition algorithms to accelerate the Krylov type solution for the Stokes-Darcy system. We use particular solutions of this system on a circular geometry to analyze the iteration operator mode by mode. We introduce a new optimization strategy of the so-called Robin parameters based on a specific linear relation between these parameters, using the min-max and the expectation minimization approaches. Moreover, we use a Krylov solver to deal with the iteration operator and accelerate this new optimized domain decomposition algorithm. Several numerical experiments are provided to validate the effectiveness of this new method. [ABSTRACT FROM AUTHOR]

This paper presents a recently developed particle simulation code package PIFE-PIC, which is a novel three-dimensional (3-D) parallel immersed finite element (IFE) particle-in-cell (PIC) simulation model for particle simulations of plasma-material interactions. This framework is based on the recently developed nonhomogeneous electrostatic IFE-PIC algorithm, which is designed to handle complex plasma-material interface conditions associated with irregular geometries using a Cartesian mesh-based PIC. Three-dimensional domain decomposition is utilized for both the electrostatic field solver with IFE and the particle operations in PIC to distribute the computation among multiple processors. A simulation of the orbital motion-limited (OML) sheath of a dielectric sphere immersed in a stationary plasma is carried out to validate parallel IFE-PIC and profile the parallel performance of the code package. Furthermore, a large-scale simulation of plasma charging at a lunar crater containing 2 million PIC cells (10 million FE/IFE cells) and about 1 billion particles, running for 20,000 PIC steps in about 154 wall-clock hours, is presented to demonstrate the high-performance computing capability of PIFE-PIC. [ABSTRACT FROM AUTHOR]

We consider in this paper numerical approximations of a phase field model for twophase ferrofluids, which consists of the Navier--Stokes equations, the Cahn--Hilliard equation, the magnetostatic equations, and the magnetic field equation. By combining the projection method for the Navier--Stokes equations and some subtle implicit-explicit treatments for coupled nonlinear terms, we construct a linear, decoupled, fully discrete finite element scheme to solve the highly nonlinear and coupled multiphysics system efficiently. The scheme is provably unconditionally energy stable and leads to a series of decoupled linear equations to solve at each time step. Through numerous numerical examples in simulating benchmark problems such as the Rosensweig instability and droplet deformation, we demonstrate the stability and accuracy of the numerical scheme. [ABSTRACT FROM AUTHOR]

The purpose of this article is to propose and analyze a new coupled multiphysics model and a decoupled stabilized finite element method for the closed-loop geothermal system, which mainly consists of a network of underground heat exchange pipelines to extract the geothermal heat from the geothermal reservoir. The new mathematical model considers the heat transfer between two different flow regions, namely the porous media flow in the geothermal reservoir and the free flow in the pipes. Darcy’s law and Navier–Stokes equations are considered to govern the flows in these two regions, respectively, while the heat equation is coupled with the flow equations to describe the heat transfer in both regions. Furthermore, on the interface between the two regions, four physically valid interface conditions are considered to describe the continuity of the temperature and the heat flux as well as the no-fluid-communication feature of the closed-loop geothermal system. In the variational formulation, an interface stabilization term with a penalty parameter is added to overcome the difficulty of the possible numerical instability arising from the interface conditions in the finite element discretization. To solve the proposed model accurately and efficiently, we develop a stabilized decoupled finite element method which decouples not only the two flow regions but also the heat field and the flow field in each region. The stability of the proposed method is proved. Four numerical experiments are provided to demonstrate the applicability of the proposed model and the accuracy of the numerical method. [ABSTRACT FROM AUTHOR]

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 CahnHilliard-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. [ABSTRACT FROM AUTHOR]

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 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 uid 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel. This article discusses a finite element splitting extrapolation for second order hyperbolic equations with time-dependent coefficients. This method possesses a higher degree of parallelism, less computational complexity, and more flexibility than Richardson extrapolation while achieving the same accuracy. By means of domain decomposition and isoparametric mapping, some grid parameters are chosen according to the problem. The multiparameter asymptotic expansion of the d-quadratic finite element error is also established. The splitting extrapolation formulas are developed from this expansion. An approximation with higher accuracy on a globally fine grid can be computed by solving a set of smaller discrete subproblems on different coarser grids in parallel. Some a posteriori error estimates are also provided. Numerical examples show that this method is efficient for solving discontinuous problems and nonlinear hyperbolic equations. [ABSTRACT FROM AUTHOR]