Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X_H and X_h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ? H, respectively, and a finite element space M_h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H⁵. If we choose H = O(h^2/5), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.     

Steady magnetohydrodynamic flow in a diverging channel with suction or blowing

An analysis is made of steady two-dimensional divergent flow of an electrically conducting incompressible viscous fluid in a channel formed by two non-parallel walls, the flow being caused by a source of fluid volume at the intersection of the walls. The fluid is permeated by a magnetic field produced by an electric current along the line of intersection of the channel walls. The walls are porous and subjected to either suction (k > 0) or blowing (k < 0) of equal magnitude on both the walls. It is found that when the Reynolds number for the flow is large and the magnetic Reynolds number is very small, boundary layers are formed on the channel walls such that a sufficient condition for the existence of a unique boundary layer solution (without separation) in the case of suction is N > 2, N being the magnetic parameter. When k = 0, boundary layer exists without separation only when N > 2. Further, it is found that the necessary and sufficient condition for the existence of a unique solution for boundary layer flow (without separation) even in the presence of blowing (k < 0) is N > 2. For given value of k, velocity at a point increases with increase in N. It is also shown that when N > 2, blowing makes the boundary layer thinner. A similarity solution for steady temperature distribution in the divergent flow is also presented when the channel walls are held at variable temperature. It is found that for fixed value of wall suction, temperature at a point decreases with increase in N. It is further shown that when N > 2, steady distribution of temperature exists even in the case of blowing at the walls.     

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.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Error estimates for two‐level penalty finite volume method for the stationary Navier–Stokes equations

Two‐level penalty finite volume method for the stationary Navier–Stokes equations based on the P₁ ? P₀ element is considered in this paper. The method involves solving one small penalty Navier–Stokes problem on a coarse mesh with mesh size H = ?^{1 / 4}h^{1 / 2}, a large penalty Stokes problem on a fine mesh with mesh size h, where 0 < ? < 1 is a penalty parameter. The method we study provides an approximate solution with the convergence rate of same order as the penalty finite volume solution (u_?h,p_?h), which involves solving one large penalty Navier–Stokes problem on a fine mesh with the same mesh size h. However, our method can save a large amount of computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Carleson Type Measures for Harmonic Mixed Norm Spaces with Application to Toeplitz Operators

Let Ω be a bounded domain in Rnwith a smooth boundary, and let h p,q be the space of all harmonic functions with a finite mixed norm. The authors first obtain an equivalent norm on h p,q, with which the definition of Carleson type measures for h p,q is obtained. And also, the authors obtain the boundedness of the Bergman projection on h p,q which turns out the dual space of h p,q. As an application, the authors characterize the boundedness(and compactness) of Toeplitz operators T μ on h p,q for those positive finite Borel measures μ.     

Nano boundary layer equation with nonlinear Navier boundary condition

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.     

A defect-correction stabilized finite element method for Navier–Stokes equations with friction boundary conditions

In this paper, we consider a defect-correction stabilized finite element method for incompressible Navier–Stokes equations with friction boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. In the defect step, an artificial viscosity parameter σ is added to the Reynolds number as a stability factor, and the Oseen iterative scheme is applied in the correction step. $H^1\times L^2$ error estimations are derived for the one-step defect-correction stabilized finite element method. In the end, some numerical results are presented to verify the theoretical analysis.     

A priori error analysis of the mortar finite element method for variational inequalities

The mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H²(D), then the mortar element solution can achieve the same order error estimate as the conforming P₁ finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Interactive computing methods for aeroelastic analysis of turbomachinery

An interaction viscous‐inviscid method for efficiently computing steady and unsteady viscous flows is presented. The inviscid domain is modeled using a finite element discretization of the full potential equation. The viscous region is modeled using a finite difference boundary layer technique. The two regions are simultaneously coupled using the transpiration approach. A time linearization technique is applied to this interactive method. For unsteady flows, the fluid is assumed to be composed of a mean or steady flow plus a harmonically varying small unsteady disturbance. Numerically exact nonreflecting boundary conditions are used for the far field conditions. Results for some steady and unsteady, laminar and turbulent flow problems are compared to linearized Navier‐Stokes or time‐marching boundary layer methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

Steady laminar boundary layer of a viscous incompressible fluid in a convergent channel with distributed suction at the wall

In this paper an attempt has been made to find the solution of the boundary layer equations for two-dimensional laminar steady motion of a viscous incompressible fluid in a convergent channel (sink flow) with suction at the wall. Suction velocity v₀ (x) ～ 1/x has been imposed at the wall and an approximate solution has been obtained with the help of similarity transformation. A solution valid at a large distance from the wall and a series solution valid near the wall have been obtained and the two solutions have been joined at a suitable point. It is seen that the boundary layer thickness diminishes as the value of the suction parameter\(\lambda ( = v_0 x/\sqrt {u_1 v} )\) increases. The velocity profile and the boundary layer parameters for solid wall (λ = 0) obtained from this solution are found to be in close agreement with the profile and the parameters calculated from the known exact solution for the solid wall problem.     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016     

Adaptive p-version of the finite element method for solving boundary value problems for ordinary second-order differential equations

We suggest an adaptive strategy for constructing a hierarchical basis for a p-version of the finite element method used to solve boundary value problems for second-order ordinary differential equations. The choice of the order of an element on each grid interval is based on estimates of the change, in the norm of C, of the approximate solution or the value of the functional to be minimized when increasing the degree of the basis function added on this interval. The results of numerical experiments estimating the method efficiency are given for sample problems whose solutions have singularities of the boundary layer type. We make a comparison with the p-version of the finite element method, which uses a uniform growth of the degree of the basis functions, and with the h-version, which uses uniform grid refinement along with an adaptive grid refinement and coarsening strategy.     

Analysis of newton multilevel stabilized finite volume method for the three‐dimensional stationary Navier‐Stokes equations

This article proposes and analyzes a multilevel stabilized finite volume method(FVM) for the three‐dimensional stationary Navier–Stokes equations approximated by the lowest equal‐order finite element pairs. The method combines the new stabilized FVM with the multilevel discretization under the assumption of the uniqueness condition. The multilevel stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multilevel‐stabilized FVM provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h_j ～ h_j‐1², j = 1,…,J. Therefore, the multilevel stabilized FVM is more efficient than the standard one‐level‐stabilized FVM. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Time-discretized steady compressible Navier–Stokes equations with inflow and outflow boundaries

The time-discretized steady compressible Navier–Stokes equations in n-dimensional bounded domains with the velocity specified only at the inflow boundary are considered. The existence and uniqueness of L ^p solutions are proved for p > n. For time-discretized steady flows, results of Kweon and Kellogg and of Kweon and Song are extended in a manner that allows for more general domains and for density-dependent viscosity coefficients. Moreover, we only require p > n which is a critical barrier in the previous works.     

Struktur der Grenzschicht zwischen einem feldfreien Plasma und einem Magnetfeld im Vakuum

Self-consistent solutions of Vlasov's equations describing a plane boundary layer without shear are given. The currents within the layer are due to free and trapped electrons. The ion current is neglected sincem _e ?m _i . The free electrons and the ions are assumed Maxwellian at infinity. For the trapped electrons a one parameter family of distribution functions is assumed. Profiles are computed for potentials, fields, densities and currents for different numbers of trapped electrons. Numerical solutions are obtained by applying the method of finite elements to a variational formulation of the problem.