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In this paper, we design a decoupled scheme for solving a fluid–fluid interaction problem, which includes two Navier–Stokes equations coupled by some nonlinear interface conditions. Compared with two decoupled schemes proposed by Connors et al. (2012, Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal. , 50 , 1297–1319) for the fluid–fluid interaction problem, we deal with these nonlinear interface conditions by applying explicit scheme. The presented fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization and explicit treatment for the interface conditions and the nonlinear terms. Moreover, the unconditional energy stability is established and error estimate for the fully discrete scheme is also showed. Finally, some numerical experiments are provided to verify the theoretical results, which illustrate the accuracy and efficiency of the presented scheme. [ABSTRACT FROM AUTHOR]

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this article, we study a multi-physics model for the closed-loop geothermal system consisting of some pipelines and a reservoir. The mathematical model for simulating this system mainly focuses on heat transfer between porous media in the reservoir and fluid flow in the pipelines. Besides, to describe the fluid and heat transfer in two regions, the governing equations of the mathematical model are denoted by the steady Navier-Stokes/Darcy equations coupled with the convection-diffusion equations. Next, we design a perturbation system to prove the well-posedness of the governing equations. Moreover, in variational formulation, a Nitsche's penalty term is added to overcome the difficulty of instability caused by the interface condition of heat flux. Furthermore, a decoupled iterative algorithm is proposed to solve the considered equations based on the finite element method, and the stability and convergence of the presented algorithm are shown. Finally, some numerical tests are performed to verify the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness and H 2 -regularity of solution u for a semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity under the stability and uniqueness conditions. Moreover, we design the finite element method based on P 1 -element for the semi-linear elliptical equation and prove the existence and uniqueness of the finite element solution u h by the finite element Oseen iterative method under the stability and uniqueness condition. Next, we prove the existence and uniqueness of the solution u for the semi-linear elliptical equation by the finite element solution u h under the stability and uniqueness conditions. Furthermore, we provide the H 1 − L 2 optimal error estimates of the finite element solution u h with respect to the exact solution u for the semi-linear elliptical equation under the stability and uniqueness conditions. Finally, some numerical tests are presented to show the efficiency of the proposed method for the semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel fully discrete semi-implicit stabilized finite element method for the Cahn-Hilliard-Navier-Stokes phase-field model by using the lowest equal-order (P 1 / P 1 / P 1 / P 1) finite element pair, which consists of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the first order semi-implicit scheme combined with convex splitting approximation for the temporal discretization. We prove that the fully discrete scheme is unconditional energy stable and mass conservative. We also carry out optimal error estimates both in time and space for the phase function, chemical potential and velocity in the appropriate norms. Finally, several numerical experiments are presented to confirm the theoretical results and the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we investigate fully discrete schemes for the Allen-Cahn and Cahn-Hilliard equations respectively, which consist of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the semi-implicit scheme for the temporal discretization. With reasonable stability conditions, it is shown that the proposed schemes are energy stable. Furthermore, by defining a new projection operator, we deduce the optimal error estimates. Some numerical experiments are presented to confirm the theoretical predictions and the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-level nonconforming rotated finite element (TNRFE) method for solving the Navier–Stokes equations. A new nonconforming rotated finite element (NRFE) method was proposed by Douglas added by conforming bubbles to velocity and discontinuous piecewise constant to the pressure on quadrilateral elements possessing favorable stability properties. The TNRFE method involves solving a small Navier–Stokes problem on a coarse mesh with mesh size H and a large linearized Navier–Stokes problem on a fine mesh with mesh size h by the NRFE method. If we choose h = O H 2 , the TNRFE method gives the convergence rate of the same order as that of the NRFE method. Compared with the NRFE method, the TNRFE method can save a large amount of CPU time. In this paper, the stability of the approximate solutions and the error estimates are proved. Finally, the numerical experiments are given, and results indicate that the method is practicable and effective. [ABSTRACT FROM AUTHOR]

This paper proposes a deferred defect‐correction method for the time‐dependent nonlinear Darcy‐Brinkman model based on the mixed finite element method. The presented method combines the deferred correction strategy with the defect‐correction method and involves two steps. In the first step, a defect Darcy‐Brinkman model is solved. Then in the second step, a correction Darcy‐Brinkman model based on the deferred correction approach is solved. Moreover, unconditional stability and convergence of the presented method are deduced. Finally, several numerical examples are given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a decoupled modified characteristic finite element method with different subdomain time steps for the mixed stabilized formulation of nonstationary dual-porosity-Navier-Stokes model. Based on partitioned time-stepping methods, the mixed system with a stabilization term is decoupled, which means that the Navier-Stokes equations and two different Darcy equations are solved independently at each time step of subdomains. In particular, we solve the Navier-Stokes equations by the modified characteristic finite element method, which overcomes the computational inefficiency caused by the nonlinear term. In order to increase the efficiency, different time steps are used to different subdomains. We prove the error convergence of solutions by mathematical induction, whose proof implies the uniform L ∞ -boundedness of the fully discrete velocity solution in conduit flow. Finally, some numerical tests are presented to show efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, based on interior penalty discontinuous Galerkin method and mixed finite element method we consider a strongly conservative discretization for the rearranged Stokes–Biot model. Theoretical analysis and numerical modeling are carried out. From the view of mathematics, we firstly present the existence and uniqueness of solution of the numerical scheme. Then, the analysis of stability and priori error estimates are derived. From the point of numerical simulation, we report the numerical examples under uniform meshes, which well validate the analysis of convergence and the strong mass conservation. [ABSTRACT FROM AUTHOR]

A decoupled finite element method with different time steps for the nonstationary Darcy--Brinkman problem is considered in this paper. Moreover, for the presented method, the stability analysis and error estimates are deduced. Finally, numerical tests are provided that demonstrate the efficiency of the method. It is found the presented method can save lots of computational time compared with standard coupled method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity. Firstly, we prove the existence and uniqueness of solution for its variational problem. Then, we obtain the stability and convergence of the corresponding finite element Galerkin approximation. Furthermore, we derive the optimal error estimates in the L 2 and H 1 norms for the finite element approximations respectively. Finally, several numerical experiments are provided to confirm the above theoretical results. [ABSTRACT FROM AUTHOR]

A new multiscale finite element method for the two-dimensional (2D) transient Navier–Stokes equations is proposed in this paper. This new method is based on multiscale enrichment with the lowest equal-order finite element pair. Under certain regularity assumptions, the optimal error estimates in -norm for velocity and -norm for pressure are obtained. Especially, via applying a new dual problem for the transient Navier–Stokes problem and some techniques in the proof process, we establish the convergence of the optimal order in -norm for the velocity. Finally, a numerical example confirms our theory analysis and validates the high effectiveness of this new method. [ABSTRACT FROM AUTHOR]

Abstract This paper considers a Crank–Nicolson extrapolation scheme based on mixed finite element method to solve the three-dimensional incompressible magnetohydrodynamics (MHD) equations. We prove that the fully discrete scheme is almost unconditionally stable and convergent, i.e., stable and convergent when the time step is less than or equal to a constant. By a new negative norm technique, the optimal error estimates in L 2 -norm are derived. Meanwhile, the numerical investigations provide a sufficient support for the theoretical analysis. [ABSTRACT FROM AUTHOR]

Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract A finite element method based on P 1 -element for the 3D semilinear elliptic boundary value problem with small viscosity is considered. The existence, uniqueness and uniform optimal H 1 and suboptimal L 2 error estimates of the finite element solution are provided, where 0 < ν < < 1 ≤ β. Also, the numerical tests are made to show the efficiency of the error estimates of the finite element solution for the 3D semilinear elliptic boundary value problem with small viscosity. [ABSTRACT FROM AUTHOR]

In this article, we consider the stationary Oldroyd fluid equations from the large time behavior research of the nonstationary equations. Thus, to obtain its numerical solution, we first solve the nonstationary Oldroyd fluid equations via the Euler implicit/explicit finite element method with the integral term discretized by the right‐hand rectangle rule, then increase the total time (i.e., number of time steps) to approximate the solution of the original stationary equations. Under a new uniqueness condition (A2), we prove the exponential stability of the solution pair { u ¯ , p ¯ } for the stationary equations and the almost unconditional stability of the numerical method. Furthermore, we also obtain the uniform optimal H 1 and L 2 error estimates in time integral 0 ≤ t < + ∞ . Finally, several numerical experiments are provided to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study stability and convergence of fully discrete finite element method on large timestep which used Crank–Nicolson extrapolation scheme for the nonstationary Navier–Stokes equations. This approach bases on a finite element approximation for the space discretization and the Crank–Nicolson extrapolation scheme for the time discretization. It reduces nonlinear equations to linear equations, thus can greatly increase the computational efficiency. We prove that this method is unconditionally stable and unconditionally convergent. Moreover, taking the negative norm technique, we derive the L 2 , H 1 -unconditionally optimal error estimates for the velocity, and the L 2 -unconditionally optimal error estimate for the pressure. Also, numerical simulations on unconditional L 2 -stability and convergent rates of this method are shown. [ABSTRACT FROM AUTHOR]

Many stable and convergent numerical methods are not pressure robust for an incompressible flow problem, due to the relaxation of a discrete divergence constraint called poor mass conservation. As a result, the velocity error is polluted by the pressure error, particularly when the fluid viscosity is relatively small and pressure is comparably large. A grad-div stabilization is an efficient remedy in conforming finite element methods. In this paper, an analogue of the grad-div stabilization that the penalization of the jumps of the normal velocities over facets of the triangulation in discontinuous Galerkin (DG) methods is investigated for a coupled Stokes and Darcy problem, it can efficiently improve the accuracy of velocity and mass conservation. Firstly, we present the existence and uniqueness of a solution, unconditional stability and a priori estimates for the DG method. Then, by using the principle of superposition and orthogonal decomposition cleverly, we prove the limiting behavior that the DG method with an analogue of grad-div stabilization for a coupled Stokes and Darcy problem converge to the strongly conservative BDM DG method with a convergence rate of O (λ f − 1 + λ s − 1) , it allows for arbitrarily large stabilization parameters. Finally, some numerical tests are given to validate the optimal convergence rates and their effectiveness. [ABSTRACT FROM AUTHOR]

This survey enfolds rigorous analysis of the defect-correction finite element (FE) method for the time-dependent conduction-convection problem which based on the Crank-Nicolson scheme. The method consists of two steps: solve a nonlinear problem with an added artificial viscosity term on a FE grid and correct the solutions on the same grid using a linearized defect-correction technique. The stability and optimal error estimate of the fully discrete scheme are derived. As a consequence, the effectiveness of the method to deal with high Reynolds number is illustrated in several numerical experiments. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 681-703, 2017 [ABSTRACT FROM AUTHOR]

A fully discrete second-order decoupled implicit/explicit method is proposed for solving 3D primitive equations of ocean in the case of Dirichlet boundary conditions on the side, where a second-order decoupled implicit/explicit scheme is used for time discretization, and a finite element method based on the P1( P1) − P1− P1( P1) elements for velocity, pressure and density is used for spatial discretization of these primitive equations. Optimal H1− L2− H1 error estimates for numerical solution [ABSTRACT FROM AUTHOR]

An efficient method is examined for the Oldroyd fluid which belongs to nonlinear integro-differential equations. This approach is approximately discretized by the FEM (i.e. finite element method) in space level, and a second-order Crank–Nicolson extrapolation in time. It reduces nonlinear equations to linear equations. Thus can greatly increase the computational efficiency. In order to approximate the integral term, we apply the "midpoint rule" approach. The error of the integral term to the midpoint rule is determined. We establish that the method is unconditionally stable and convergent. The L 2 -optimal error estimate with second order accuracy in time and space level is obtained without imposing any restriction on time-step size, while all previous works of higher-order scheme require certain time-step conditions. Moreover, corresponding numerical experiments are shown to demonstrate the accuracy, the unconditional stability and convergence of this method for the Oldroyd fluid. The structure is also suitable for the Navier–Stokes equations (i.e. ρ = 0 in the Oldroyd fluid). • Unconditional convergence is firstly established to higher-order temporal scheme. • The frame established can be extended to the Navier–Stokes related equations. • Nonlinear and integral issue are tackled by extrapolation scheme and midpoint rule. [ABSTRACT FROM AUTHOR]

A second order time–space implicit/explicit iterative scheme for the stationary Navier–Stokes equations is designed, where the spatial discretization is based on the mixed finite element method and the time discretization is based on the second order implicit/explicit (the Crank–Nicolson/Admas–Bashforth) scheme. Under a weak uniqueness condition, the optimal H 1 - L 2 error estimates related to the mesh size h and time step size τ of the iterative solution ( u h n , p h n ) to the exact solution ( u ̃ , p ̃ ) and the optimal L 2 error estimate related to h and τ of the iterative solution u h n to the exact solution u ̃ are provided. In numerical aspect, some comparisons with the first order time–space iterative method are made to confirm the efficiency of the proposed second order scheme. [ABSTRACT FROM AUTHOR]

The paper is concerned with the adaptive linear finite element solution of linear one-dimensional elliptic differential equations using equidistributing meshes. A strategy is developed for defining such meshes based on a residual-based a posteriori error estimate. The mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation (i.e., the mesh equation). An iterative algorithm is proposed for solving this system for the mesh and the finite element solution. The existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh, which is also established in the current work. Error bounds for the finite element solution are obtained for the equidistributing and quasi-equidistributing meshes. They show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes. The departure from the equidistributing meshes has only a mild effect on the finite element error. Numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper proposes a second-order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank-Nicolson/Adams-Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for L2, H1 error estimates of the velocity and L2 error estimates of the pressure at different values of viscoelastic viscosities α, different values of relaxation time λ1, different values of null viscosity coefficient μ0 are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity μ and different relaxation time λ1 are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A new two-step stabilized finite element method for the 2D/3D stationary Navier-Stokes equations based on local Gauss integration is introduced and analyzed in this paper. The method consists of solving one Navier-Stokes problem based on the P− P finite element pair and then solving a general Stokes problem based on the P− P finite element pair, i.e., computes a lower order predictor and a higher order corrector. Moreover, the stability and convergence of the present method are deduced, which show that the new method provides an approximate solution with the convergence rate of the same order as the P− P stabilized finite element solution solving the Navier-Stokes equations on the same mesh width. However, our method can save a large amount of computational time. Finally, numerical tests confirm the theoretical results of the method. [ABSTRACT FROM AUTHOR]

In this article, a decoupled fully discrete scheme for solving 2D magnetohydrodynamics (MHD) equations is proposed. The decoupled scheme is used for time discretization, and the finite element method is used for spatial discretization. Firstly, the almost unconditional stability ( Δ t ≤ C ) of this scheme is established. Then optimal L 2 and H 1 error estimates of numerical solution are provided. Finally, a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme. [ABSTRACT FROM AUTHOR]

In this paper, a modified method of characteristics mixed defect-correction finite element method (MMOCMDCFEM) for the time-dependent Navier–Stokes problems, which is led by combining the characteristics time discretization with the two-step defect-correction in space, is presented. In this method, the hyperbolic part (the temporal and advection term) are treated by a characteristic tracking scheme. Then, we solve the equations with an added artificial viscosity term and correct these solutions by using a defect-correction technique. The theory analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCMDCFEM, we first present some numerical results of analytical solution problems. Then, we give some numerical results of lid-driven cavity flow withand. We also give some numerical results of flow around a cylinder problems, and compare the results with MMOCFEM’s. The numerical results show that this method is highly Efficient. [ABSTRACT FROM PUBLISHER]

Some finite element iterative methods related to viscosities are designed to solve numerically the steady 2D/3D Navier–Stokes equations. The two-level finite element iterative methods are designed to solve numerically the steady 2D/3D Navier–Stokes equations for a large viscosity ν such that a strong uniqueness condition holds. The two-level finite element iterative methods consist of using the Stokes, Newton and Oseen iterations of m times on a coarse mesh with mesh size H and computing the Stokes, Newton and Oseen correction of one time on a fine grid with mesh size h ≪ H . Moreover, the one-level Oseen finite element iterative method based on a fine mesh with a small mesh size is designed to solve numerically the steady 2D/3D Navier–Stokes equations for small viscosity ν such that a weak uniqueness condition holds. The uniform stability and convergence of these methods with respect to ν and mesh sizes h and H and iterative times m are provided. [ABSTRACT FROM AUTHOR]

In this article, a streamline diffusion finite element method is proposed and analyzed for stationary incompressible magnetohydrodynamics (MHD) equations. This method is stable for any combinations of velocity, pressure, and magnet finite element spaces, without requiring Ladyzenskaja-Babuška-Brezzi (LBB) condition. The well-posedness and convergence (at optimal error rate) of this scheme are proved in terms of some conditions. Two numerical experiments are illustrated to validate our theoretical analysis and show the streamline diffusion finite element approach is effective for solving the MHD problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1877-1901, 2014 [ABSTRACT FROM AUTHOR]

This paper considers finite element approximation and three coupled type iterations of stationary magnetohydrodynamics (MHD) equations on a general Lipschitz domain. An additional Lagrange multiplier r is introduced related to divergence free of magnetic field b and the b is analyzed in H ( curl ; Ω ) space, which is proposed in Schötzau (2004). In finite element discretization, the hydrodynamic unknowns are approximated by stable finite element pairs, and the magnetic unknown is discretized by curl-conforming Nédélec element spaces. The well-posedness of this formula and the optimal error estimate are provided. Based on this, for numerical implementation of this scheme, we propose and discuss three coupled type iterative methods which are stable and convergent under different conditions. Specifically, Iteration I is stable and convergent under strong condition. Iteration II is stable and convergent under weaker condition. Iteration III is unconditionally stable and convergent under the weakest condition. Finally, some numerical tests confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]