Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.

LEGENDRE SPECTRAL-FINITE ELEMENT METHOD FOR THE THREE-DIMENSIONAL UNSTEADY NAVIER-STOKES EQUATION

This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artificial compression. The generalized stability and convergence are proved strictly on the assumption that the two-dimensional inf-sup condition of the finite element approximation is satisfied.

TWO-LEVEL METHOD FOR UNSTEADY NAVIER-STOKES EQUATIONS IN STREAM FUNCTION FORM

Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied.This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid.Moreover,the scaling between these two grid sizes is super-linear.Approximation,stability and convergence aspects of a fully discrete scheme are analyzed.At last a numrical example is given whose results show that the algorithm proposed in this paper is effcient.

Error Analysis of the Nonconforming P1 Finite Element Method to the Sequential Regularization Formulation for Unsteady Navier-Stokes Equations

In this paper we investigate the nonconforming P_(1) finite element ap-proximation to the sequential regularization method for unsteady Navier-Stokes equations.We provide error estimates for a full discretization scheme.Typi-cally,conforming P_(1) finite element methods lead to error bounds that depend inversely on the penalty parameter ∈.We obtain an ∈-uniform error bound by utilizing the nonconforming P_(1) finite element method in this paper.Numerical examples are given to verify theoretical results.

From Hölder Continuous Solutions of 3D Incompressible Navier-Stokes Equations to No-Finite Time Blowup on [ 0,∞ ]

This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.

On the stability of shear flows of Prandtl type for the steady Navier-Stokes equations

In this paper, we study the stability of shear flows of Prandtl type as(U(y/√ν), 0) for the steady Navier-Stokes equations under a natural spectral assumption on the linearized NS operator. The key ingredient is to solve the Orr-Sommerfeld equation. For this, we develop a direct energy method combined with the compactness method, which may be of independent interest.

FOURIER-LEGENDRE SPECTRAL METHOD FOR THE UNSTEADY NAVIER-STOKES EQUATIONS

Fourier-Legendre spectral approximation for the unsteady Navier-Stokes equations is analyzed. The generalized stability and convergence are proved respectively.

THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA

A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and regularity of the constructed solution.The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables.With these finer estimates,we obtain higher order regularity of the constructed solution to Navier-Stokes equation,so that all of the derivatives in the equation of conservative form are in the strong sense.Moreover,this regularity also allows us to identify a function space such that the stability of the solutions can be established there,which eventually implies the uniqueness.

ON LOCAL CONTROLLABILITY FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY DEPENDENT VISCOSITIES

In this paper,we study the controllability of compressible Navier-Stokes equations with density dependent viscosities.For when the shear viscosityμis a positive constant and the bulk viscosityλis a function of the density,it is proven that the system is exactly locally controllable to a constant target trajectory by using boundary control functions.

Analysis and Numerical Computations of the Multi-Dimensional,Time-Fractional Model of Navier-Stokes Equation with a New Integral Transformation

The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.

GLOBAL SOLUTIONS TO THE 2D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SOME LARGE INITIAL DATA

We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part Pu_0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.

THE LOW MACH NUMBER LIMIT FOR ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH A REVISED MAXWELL'S LAW

We investigate the low Mach number limit for the isentropic compressible NavierStokes equations with a revised Maxwell's law(with Galilean invariance) in R^(3). By applying the uniform estimates of the error system, it is proven that the solutions of the isentropic Navier-Stokes equations with a revised Maxwell's law converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.

A FULL DISCRETE STABILIZED METHOD FOR THE OPTIMAL CONTROL OF THE UNSTEADY NAVIER-STOKES EQUATIONS

In this paper, a full discrete local projection stabilized （LPS） method is proposed to solve the optimal control problems of the unsteady Navier-Stokes equations with equal order elements. Convective effects and pressure are both stabilized by using the LPS method. A priori error estimates uniformly with respect to the Reynolds number are obtained, providing the true solutions are sufficient smooth. Numerical experiments are implemented to illustrate and confirm our theoretical analysis.

Linear System Solutions of the Navier-Stokes Equations with Application to Flow over a Backward-Facing Step

Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.

UNSTEADY/STEADY NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON ARTIFICIAL COMPRESSIBILITY

A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.

RESIDUAL A POSTERIORI ERROR ESTIMATE OF A NEW TWO-LEVEL METHOD FOR STEADY NAVIER-STOKES EQUATIONS

Residual-based a posteriori error estimate for conforming finite element solutions of incompressible Navier-Stokes equations, which is computed with a new two-level method that is different from Volker John, is derived. A posteriori error estimate contains additional terms in comparison to the estimate for the solution obtained by the standard finite element method. The importance of the additional terms in the error estimates is investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than the convergence of discrete solution. The two-level method aims to solve the nonlinear problem on a coarse grid with less computational work, then to solve the linear problem on a fine grid, which is superior to the usual finite element method solving a similar nonlinear problem on the fine grid.

Physics-Driven Learning of the Steady Navier-Stokes Equations using Deep Convolutional Neural Networks

Recently,physics-driven deep learning methods have shown particular promise for the prediction of physical fields,especially to reduce the dependency on large amounts of pre-computed training data.In this work,we target the physicsdriven learning of complex flow fields with high resolutions.We propose the use of Convolutional neural networks(CNN)based U-net architectures to efficiently represent and reconstruct the input and output fields,respectively.By introducingNavier-Stokes equations and boundary conditions into loss functions,the physics-driven CNN is designed to predict corresponding steady flow fields directly.In particular,this prevents many of the difficulties associated with approaches employing fully connected neural networks.Several numerical experiments are conducted to investigate the behavior of the CNN approach,and the results indicate that a first-order accuracy has been achieved.Specifically for the case of a flow around a cylinder,different flow regimes can be learned and the adhered“twin-vortices”are predicted correctly.The numerical results also show that the training for multiple cases is accelerated significantly,especially for the difficult cases at low Reynolds numbers,and when limited reference solutions are used as supplementary learning targets.

A IP_N×IP_N Spectral Element Projection Method for the Unsteady Incompressible Navier-Stokes Equations

In this paper,we present a IP_N×IP_N spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equa- tions.The main purpose of this work consists of:(i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral el- ement approaches in space;(ii) construction of a stable IP_N×IP_N method together with a IP_N→IP_(N-2) post-filtering.The link of different methods will be clarified.The key feature of our method lies in that only one grid is needed for both velocity and pressure variables,which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis,the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.

Accuracy Analysis of the Boundary Integral Method for Steady Navier-Stokes Equations around a Rotating Obstacle

This paper deals with the boundary integral method to study the Navier-Stokes equations around a rotating obstacle. The detail of this method is that the exterior domain is truncated into a bounded domain and a new exterior domain by introducing some open ball BR, and the nonlinear problem in the bounded domain and the linearized problem in the new exterior domain are considered and the approximation coupled problem is obtained. We show that the error between the solution u of Navier-Stokes equations around a rotating obstacle and the solution ue of the approximation coupled problem is O（R-1/4） in the Hl-seminorm when Iwl does not exceed some constant.

Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

This paper presents a Martingale regularization method for the stochas-tic Navier–Stokes equations with additive noise.The original system is split into two equivalent parts,the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier–Stokes equations with relatively-higher regular-ities.Meanwhile,a fractional Laplace operator is introduced to regularize the noise term.The stability and convergence of numerical scheme for the pathwise modified Navier–Stokes equations are proved.The comparisons of non-regularized and reg-ularized noises for the Navier–Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.