Numerical analysis of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D incompressible flow with large data

In this article, we develop a branch of nonsingular solutions of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D stationary Navier‐Stokes equations without relying on the unique solution condition. The method presented consists of capturing almost all information of initial problem (the nonlinear problems) on the coarsest mesh and then performs one Picard defect correction (the linear problems) on each subsequent mesh based on previous information thus only solving one large linear systems. What is more, the method presented can results in a better coefficient matrix in the model presented with small viscosity. Theoretical results show that the method presented is derived with the convergence rate of the same order as the corresponding finite volume method/finite element method solving the stationary Navier‐Stokes equations on a fine mesh. Therefore, the method presented is definitely more efficient than the standard finite volume method/finite element method. Finally, numerical experiments clearly show the efficiency of the method presented for solving the stationary Navier‐Stokes equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 30–50, 2018     

Eigenvalue estimates for a preconditioned Galerkin matrix arising from mixed finite element discretizations of viscous incompressible flows

The article deals with Galerkin matrices arising with finite element discretizations of the Navier–Stokes system. Usually these matrices are indefinite and nonsymmetric. They have to be preconditioned if a related linear system is to be solved efficiently by an iterative method. We consider preconditioning by a pressure mass matrix. It is shown how upper and lower bounds of the eigenvalues of a preconditioned Galerkin matrix may be found by variational arguments.     

A parallel two-level finite element variational multiscale method for the Navier–Stokes equations

A combination method of two-grid discretization approach with a recent finite element variational multiscale algorithm for simulation of the incompressible Navier–Stokes equations is proposed and analyzed. The method consists of a global small-scale nonlinear Navier–Stokes problem on a coarse grid and local linearized residual problems in overlapped fine grid subdomains, where the numerical form of the Navier–Stokes equations on the coarse grid is stabilized by a stabilization term based on two local Gauss integrations at element level and defined by the difference between a consistent and an under-integrated matrix involving the gradient of velocity. By the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method.     

A new relaxed HSS preconditioner for saddle point problems

We present a preconditioner for saddle point problems. The proposed preconditioner is extracted from a stationary iterative method which is convergent under a mild condition. Some properties of the preconditioner as well as the eigenvalues distribution of the preconditioned matrix are presented. The preconditioned system is solved by a Krylov subspace method like restarted GMRES. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioner.     

Explicit solution of the incompressible Navier–Stokes equations with linear finite elements

A three-field finite element scheme for the explicit iterative solution of the stationary incompressible Navier–Stokes equations is studied. In linearized form the scheme is associated with a generalized time-dependent Stokes system discretized in time. The resulting system of equations allows for a stable approximation of velocity, pressure and stress deviator tensor, by means of continuous piecewise linear finite elements, in both two- and three-dimensional space. Convergence in an appropriate sense applying to this finite element discretization is demonstrated, for the stationary Stokes system.     

A Two‐Parameter Stabilized Finite Element Method for Incompressible Flows

Based on two‐grid discretizations, a two‐parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher‐order finite element interpolants of the velocity into a lower‐order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017     

Existence and zero‐electron‐mass limit of strong solutions to the stationary compressible Navier‐Stokes‐Poisson equation with large external force

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier‐Stokes‐Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero‐electron‐mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier‐Stokes equations with large forces, a system of stationary compressible Navier‐Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier‐Stokes equation, linear compressible Navier‐Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero‐electron‐mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier‐Stokes‐Poisson equations.     

Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

In this paper, we compare two block triangular preconditioners for different linearizations of the Rayleigh–Bénard convection problem discretized with finite element methods. The two preconditioners differ in the nested or nonnested use of a certain approximation of the Schur complement associated to the Navier–Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time. This investigation reveals some inconsistencies in the literature that are hereby discussed. We observe that the nonnested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard–Newton scheme. Numerical results of two‐ and three‐dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.     

Eigenvalue bounds for the Schur complement with a pressure convection–diffusion preconditioner in incompressible flow computations

If the stationary Navier–Stokes system or an implicit time discretization of the evolutionary Navier–Stokes system is linearized by a Picard iteration and discretized in space by a mixed finite element method, there arises a saddle point system which may be solved by a Krylov subspace method or an Uzawa type approach. For each of these resolution methods, it is necessary to precondition the Schur complement associated to the saddle point problem in question. In the work at hand, we give upper and lower bounds of the eigenvalues of this Schur complement under the assumption that it is preconditioned by a pressure convection–diffusion matrix.     

Superconvergence of discontinuous Galerkin finite element method for the stationary Navier‐Stokes equations

This article focuses on discontinuous Galerkin method for the two‐ or three‐dimensional stationary incompressible Navier‐Stokes equations. The velocity field is approximated by discontinuous locally solenoidal finite element, and the pressure is approximated by the standard conforming finite element. Then, superconvergence of nonconforming finite element approximations is applied by using least‐squares surface fitting for the stationary Navier‐Stokes equations. The method ameliorates the two noticeable disadvantages about the given finite element pair. Finally, the superconvergence result is provided under some regular assumptions. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 421–436, 2007     

Superconvergence of a stabilized finite element approximation for the Stokes equations using a local coarse mesh L2 projection

This article first recalls the results of a stabilized finite element method based on a local Gauss integration method for the stationary Stokes equations approximated by low equal‐order elements that do not satisfy the inf‐sup condition. Then, we derive general superconvergence results for this stabilized method by using a local coarse mesh L² projection. These supervergence results have three prominent features. First, they are based on a multiscale method defined for any quasi‐uniform mesh. Second, they are derived on the basis of a large sparse, symmetric positive‐definite system of linear equations for the solution of the stationary Stokes problem. Third, the finite elements used fail to satisfy the inf‐sup condition. This article combines the merits of the new stabilized method with that of the L² projection method. This projection method is of practical importance in scientific computation. Finally, a series of numerical experiments are presented to check the theoretical results obtained. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 115‐126, 2012     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A two‐level finite element method for the stationary Navier‐Stokes equations based on a stabilized local projection

In this article, we propose a two‐level finite element method to analyze the approximate solutions of the stationary Navier‐Stokes equations based on a stabilized local projection. The local projection allows to circumvent the Babuska‐Brezzi condition by using equal‐order finite element pairs. The local projection can be used to stabilize high equal‐order finite element pairs. The proposed method combines the local projection stabilization method and the two‐level method under the assumption of the uniqueness condition. The two‐level method consists of solving a nonlinear equation on the coarse mesh and solving a linear equation on fine mesh. The nonlinear equation is solved by the one‐step Newtonian iteration method. In the rest of this article, we show the error analysis of the lowest equal‐order finite element pair and provide convergence rate of approximate solutions. Furthermore, the numerical illustrations coincide with the theoretical analysis expectations. From the view of computational time, the results show that the two‐level method is effective to solve the stationary Navier‐Stokes equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Local projection FEM stabilization for the time‐dependent incompressible Navier–Stokes problem

We consider conforming finite element (FE) approximations of the time‐dependent, incompressible Navier–Stokes problem with inf‐sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence‐free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39–77 for edge‐stabilized FE approximation of the Navier–Stokes problem. Some numerical experiments complement the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1224–1250, 2015     

A penalty‐FEM for navier‐stokes type variational inequality with nonlinear damping term

In this article, we consider a penalty finite element (FE) method for incompressible Navier‐Stokes type variational inequality with nonlinear damping term. First, we establish penalty variational formulation and prove the well‐posedness and convergence of this problem. Then we show the penalty FE scheme and derive some error estimates. Finally, we give some numerical results to verify the theoretical rate of convergence. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 918–940, 2017     

An L2 finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.     

A priori and a posteriori estimates for three‐dimensional Stokes equations with nonstandard boundary conditions

In this article, we study the Stokes problem with some nonstandard boundary conditions. The variational formulation decouples into a system for the velocity and a Poisson equation for the pressure. The corresponding discrete system do not need an inf‐sup condition. Hence, the velocity is approximated with “ curl ” conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a priori and a posteriori estimates and we finally concluded with numerical tests. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Dual‐primal mixed finite elements for elliptic problems

In this article, a novel dual‐primal mixed formulation for second‐order elliptic problems is proposed and analyzed. The Poisson model problem is considered for simplicity. The method is a Petrov—Galerkin mixed formulation, which arises from the one‐element formulation of the problem and uses trial functions less regular than the test functions. Thus, the trial functions need not be continuous while the test functions must satisfy some regularity constraint. Existence and uniqueness of the solution are proved by using the abstract theory of Nicolaides for generalized saddle‐point problems. The Helmholtz Decomposition Principle is used to prove the inf‐sup conditions in both the continuous and the discrete cases. We propose a family of finite elements valid for any order, which employs piecewise polynomials and Raviart—Thomas elements. We show how the method, with this particular choice of the approximation spaces, is linked to the superposition principle, which holds for linear problems and to the standard primal and dual formulations, addressing how this can be employed for the solution of the final linear system. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 137–151, 2001     

Stability and convergence of efficient Navier‐Stokes solvers via a commutator estimate

For strong solutions of the incompressible Navier‐Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf‐sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier‐Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.     

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter \({\varepsilon}\), which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter \({\varepsilon\longrightarrow 0}\). Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.