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In this article, we consider the coupled Navier-Stokes and Darcy problem with the Beavers-Joseph interface condition. With suitable restrictions of physical parameters α and ν, we prove the existence and local uniqueness of a weak solution. Then we propose a coupled finite element scheme and a decoupled and linearized scheme based on two-grid finite element. Under suitable further restrictions, their optimal error estimates are obtained. Finally numerical experiments indicate the validity of the theoretical results as well as the efficiency and effectiveness of the decoupled and linearized two-grid algorithm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1009-1030, 2015 [ABSTRACT FROM AUTHOR]

In this article, supposing that the velocity, pressure, and temperature are approximated by the elements [ABSTRACT FROM AUTHOR]

A two-level method in space and time for the time-dependent Navier-Stokes equations is considered in this article. The approximate solution u M∈ H M is decomposed into the large eddy component v∈ H m( m < M) and the small eddy component w∈ H [ABSTRACT FROM AUTHOR]

SUMMARY An adaptive variational multiscale method for the Stokes equations is presented in this paper. We solve the coarse scale problem on the coarse mesh and approximate the fine scale solution by solving a series of local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, numerical examples are presented to verify the algorithm.Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We present a posteriori error estimate for a defect correction method for approximating solutions of the stationary conduction convection problems in two dimension. The defect correction method is aiming at small viscosity ν. A reliable a posteriori error estimation is derived for the defect correction method. Finally, two numerical examples validate our theoretical results. The first example is a problem with known solution and the second example is a physical model of square cavity stationary flow. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 [ABSTRACT FROM AUTHOR]

This report considers a variable time-stepping algorithm proposed by Dahlquist, Liniger and Nevanlinna and discusses its application to the unsteady Stokes/Darcy model. Although long-time forgotten and little explored, the algorithm performs advantages in variable time-stepping analysis of various fluid flow systems, including the coupled Stokes/Darcy model. We first prove that the approximate solutions to the unsteady Stokes/Darcy model are unconditionally stable due to the G -stability of the method. Then variable time-stepping error analysis follows from the combination of G -stability and consistency of the method. Numerical experiments verify the theoretical results, demonstrating both accuracy and stability of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the velocity error analysis of H(div)-conforming DG method for the semi-discrete time-dependent Navier–Stokes equations. Firstly, we prove that the L ∞ (0 , T ; L 2 (Ω)) error of the velocity is optimal and pressure-robust, but the constants in the velocity error bound are dependent on the inverse power of the viscosity, so it is not semi-robust. Secondly, we focus on pressure-robust and semi-robust velocity error analysis at high Reynolds numbers. By introducing the Raviart–Thomas interpolation operator, we prove that when the condition ν < C h is satisfied, the L ∞ (0 , T ; L 2 (Ω)) error of the velocity, which is pressure-robust and semi-robust, is quasi-optimal. Numerical experiments are carried out to verify the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a second-order numerical algorithm for the non-stationary mixed Stokes/Darcy equations with Beavers–Joseph–Saffman's interface condition. The scheme is based on a finite element method in space and the parareal method with spectral deferred correction in time. We present the unconditional stability and the optimal error estimate of the full-discrete scheme. Finally, some numerical experiments are given to verify the effectiveness. [ABSTRACT FROM AUTHOR]

The flow of non-Newtonian fluids is extremely important in a variety of industrial applications and processes, including painting, printing, and coating. The goal of this paper is to develop a mathematical model for an incompressible, non-isothermal third-grade fluid as it travels through a minor gap between two heated rolls that are revolving in the opposite direction. Using appropriate dimensionless parameters, the dimensionless version of the governing equations is constructed. To simplify the nondimensional form of the governing equations, the LAT (lubrication approximation theory) is applied. It is mentioned that the perturbation approach should be used. It is a very important tool in modern mathematical physics and practical fluid mechanics, and it is used a lot. Through perturbation techniques, exact solutions for velocity distribution, flow rate, temperature distribution, and pressure gradient are offered. To observe the flow pattern, streamlines are also drawn. Numerical outcomes of some substantial engineering quantities like flow rate, roll separation force, power input, and pressure distribution are found. Some outcomes are displayed graphically although others are presented in tabular form. We have observed that the third-grade parameters give a way to control velocity, pressure gradients, flow rate, and coating thickness. Further, increasing the velocities ratio values lowers the coating thickness on the web. Furthermore, the Brickman number has a significant influence on the temperature profile and has increased with the increase in the Brickman number. Both the model and the numerical calculations are in agreement with the viscous fluid model that is already present in the literature. [ABSTRACT FROM AUTHOR]

This paper develops, analyzes and tests a time-accurate partitioned method for the Stokes–Darcy equations. The method combines a time filter and Backward Euler scheme, is second order accurate and provides, at no extra complexity, an estimate of the temporal error. This approach post-processes the solutions of Backward Euler scheme by adding three lines to original codes to increase the time accuracy from first order to second order. We prove long time stability and error estimates of Backward Euler plus time filter with constant time stepsize. Moreover, we extend the approach to variable time stepsize and construct adaptive algorithms. Numerical tests show convergence of our method and support the theoretical analysis. • A new time-accurate partitioned method is developed, analyzed and tested. • This approach has the advantage of improving time accuracy and convergence order. • Long time stability and error estimates for constant time stepsize are proved. • Time adaptivity guarantees accuracy while decreasing storage and complexity. [ABSTRACT FROM AUTHOR]

In this paper, we construct a second order algorithm based on the semi-implicit spectral deferred correction method for the non-stationary coupled Navier–Stokes/Darcy equations with the Beavers–Joseph–Saffman's interface condition. We present a complete theoretical analysis to prove that this algorithm is stable and possesses second order accuracy in time. Numerical examples confirm the convergence rate and the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

In this study, based on a combination of the two-grid method and the partition of unity-based domain decomposition method, we propose a new local and parallel finite element algorithm for the elliptic boundary value problem. The proposed method has three key features: (1) it inherits the flexibility and controllability of domain decomposition based on the partition of unity; (2) global fine grid correction is replaced by solving a series of locally defined approximate residual problems with homogeneous Dirichlet boundary conditions on some finer grids; (3) a global continuous finite element solution is constructed by solving a coarse grid correction problem and by assembling all the local solutions together using the partition of unity subordinate. Under appropriate assumptions, the optimal error estimates in L 2 and the energy norms are proved by new analytical results. In addition, several numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a partition of unity parallel finite element algorithm. This algorithm localizes the global residual problem of two grid method into some parallel local sub-problems, and use a simple partition of unity to assemble all the local solutions together. An oversampling technique is used and analyzed to decrease the undesirable effect of the artificial homogeneous Dirichlet boudary condition of local sub-problems. The analysis shows the error of this algorithm decays exponentially with respect to the oversampling parameter. Specially, on a regular coarse triangulation τ with mesh size H, an oversampling of diameter $H \log (1/H)$ is sufficient to preserve the optimal convergence order. Numerical results verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

The decoupled algorithm for a fluid–fluid interaction problem was first proposed by Connors et al. (2012). Compared with the study on its long time stable which was presented in Zhang et al. (2016), we adopt a new approach to prove the stability of the decoupled method on a finite time interval (0 , T herein. By using the Brouwer's fixed point theorem, the existence of the numerical solution is proved. At the same time, by considering the error estimate between the exact solution and the numerical solution, we derive the stability of the decoupled method under the condition of Δ t + h ≤ c from the inductive method. Furthermore, we prove the uniqueness of the numerical solution when Δ t is appropriately small. The main theoretical results about the decoupled algorithm in this paper are new. [ABSTRACT FROM AUTHOR]