Vorticity–velocity-pressure and stream function-vorticity formulations for the Stokes problem

We study the Stokes problem of incompressible fluid dynamics in two and three-dimension spaces, for general bounded domains with smooth boundary. We use the vorticity–velocity-pressure formulation and introduce a new Hilbert space for the vorticity. We develop an abstract mixed formulation that gives a precise variational frame and conducts to a well-posed Stokes problem involving a new velocity–vorticity boundary condition. In the particular case of simply connected bidimensional domains with homogeneous boundary conditions, the link with the classical stream function-vorticity formulation is completely described, and we show that the vorticity–velocity-pressure formulation is a natural mathematical extension of the previous one.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

The analysis of a finite element method for the Navier–Stokes equations with compressibility

Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained. Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.     

Numerical solution of the time-dependent incompressible Navier–Stokes equations by piecewise linear finite elements

In this paper we present a new method to solve the 2D generalized Stokes problem in terms of the stream function and the vorticity. Such problem results, for instance, from the discretization of the evolutionary Stokes system. The difficulty arising from the lack of the boundary conditions for the vorticity is overcome by means of a suitable technique for uncoupling both variables. In order to apply the above technique to the Navier–Stokes equations we linearize the advective term in the vorticity transport equation as described in the development of the paper. We illustrate the good performance of our approach by means of numerical results, obtained for benchmark driven cavity problem solved with classical piecewise linear finite element.     

Application of finite element method in aeroelasticity

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil, which can rotate around the elastic axis and oscillate in the vertical direction. The numerical simulation consists of the finite element approximation of the Navier–Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations, stabilization the finite element discretization and coupling of both models is discussed. Moreover, the Reynolds averaged Navier–Stokes (RANS) system of equations together with the Spallart–Almaras turbulence model is also discussed. The computational results of aeroelastic calculations are presented and compared with the NASTRAN code solutions.     

Discrete harmonics for stream function-vorticity Stokes problem

Summary. We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity. For this problem, the classical finite element method of degree one converges only in for the norm of the vorticity. We propose to use harmonic functions to approach the vorticity along the boundary. Discrete harmonics are functions that are used in practice to derive a new numerical method. We prove that we obtain with this numerical scheme an error of order for the norm of the vorticity. Received January, 2000 / Revised version received May 15, 2001 / Published online December 18, 2001     

A stream-function–vorticity variational formulation for the exterior Stokes problem in weighted Sobolev spaces

In this paper we derive a mixed variational formulation for the exterior Stokes problem in terms of the vorticity and stream function, or the vector potential in three dimensions. The main steps are the construction of the stream function (or vector potential) and the proof of the Babu?ka–Brezzi ‘inf-sup’ condition. The two- and three-dimensional cases are treated separately because the structure of the stream function differs substantially according to the number of dimensions considered. The conclusion of this work is that if the problem is set in the weighted Sobolev spaces of Hanouzet and Giroire, the analysis of the exterior Stokes problem is quite the same as if the domain were bounded.     

Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation

Summary. In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model. Received December 10, 1999 / Revised version received November 5, 2000 / Published online August 17, 2001     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

A new algorithm for simulating viscoelastic flows accommodating piecewise linear finite elements

A three-field finite element scheme designed for solving time-dependent systems of partial differential equations governing viscoelastic flows is introduced. The linearized form of this system is a generalized time-dependent Stokes system. Once a classical time-discretization is performed, the resulting three-field system of equations allows for a stable approximation of velocity, pressure and extra stress tensor, by means of continuous piecewise linear finite elements, in both two and three dimension space. Another advantage of the new formulation is the fact that it implicitly provides an algorithm for the iterative resolution of system non-linearities, in the case of viscoelastic flows. Additionally, convergence in an appropriate sense applying to these three flow fields is demonstrated, for such generalized Stokes system. Numerical results are given in order to illustrate the performance of the new approach.     

Asymptotics for a Symmetric Equation in Price Formation

We study the existence and asymptotics for large time of the solutions to a one dimensional evolution equation with non-standard right-hand side. The right-hand side involves the derivative of the solution computed at a given point. Existence is proven through a fixed point argument. When the problem is considered in a bounded interval, it is shown that the solution decays exponentially to the stationary state. This problem is a particular case of a mean-field free boundary model proposed by Lasry and Lions on price formation and dynamic equilibria. Maria P. Gualdani is supported by the NSF Grant DMS-0807636.     

On the optimal reconstruction of a solution of the Poisson equation

We consider the problem of optimal reconstruction of a solution of the generalized Poisson equation in a bounded domain Q with homogeneous boundary conditions for the case in which the right-hand side of the equation is fuzzy. We assume that right-hand sides of the equations belong to generalized Sobolev classes and finitely many Fourier coefficients of the right-hand sides of the equations are known with some accuracy in the Euclidean metric. We find the optimal reconstruction error and construct a family of optimal reconstruction methods. The problem on the best choice of the coefficients to be measured is solved.     

Necessary and sufficient conditions for the vibrations of a bounded string with directional derivatives in the boundary conditions

We obtain a closed-form expression for the classical solutions of a mixed problem describing the forced vibrations of a bounded string for two boundary modes with directional derivatives and time-dependent coefficients. We derive necessary and sufficient conditions on the right-hand side of the equation and the initial and boundary data.     

Fluid flows through fractured porous media along Beavers–Joseph interfaces

We study a fluid flow traversing a porous medium and obeying the Darcy's law in the case when this medium is fractured in blocks by an ε  -periodic (ε>0$\epsilon >0$) distribution of fissures filled with a Stokes fluid. These two flows are coupled by a Beavers–Joseph type interface condition. The existence and uniqueness of this flow in our ε-periodic structure are proved. As the small period of the distribution shrinks to zero, we study the asymptotic behaviour of the flow when the permeability and the entire contribution on the interface of the Beavers–Joseph transfer coefficients are of unity order. We find the homogenized problem verified by the two-scale limits of the coupled velocities and pressures. It is well-posed and provides the corresponding classical homogenized problem.     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

Finite element discretization of the Navier–Stokes equations with a free capillary surface

Summary. The instationary Navier–Stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions. A stable finite element discretization is presented. The key idea is the treatment of the curvature terms by a variational formulation. In the context of a discontinuous in time space–time element discretization stability in (weak) energy norms can be proved. Numerical examples in 2 and 3 space dimensions are given. Received March 26, 1999 / Revised version received October 12, 1999 / Published online November 8, 2000     

Solution of a mixed problem with oblique derivatives in the boundary conditions for the inhomogeneous string vibration equation without continuing the data

We find closed-form recursion formulas for the unique classical solution of a mixed problem describing forced vibrations of a bounded string under two boundary modes with timedependent oblique derivatives. The formulas do not use any continuation of the problem data. We obtain conditions on the right-hand side of the equation necessary and sufficient for the well-posed global solvability of the problem.     

An inverse problem for an equation of elliptic-hyperbolic type with a nonlocal boundary condition

We examine a mixed type equation with the Lavrent??ev-Bitsadze operator and an unknown right-hand side in a rectangle and study a nonlocal boundary value problem in which the values of the stream function on the lateral sides of the rectangle are equal. The solution is given as a sum of a biorthogonal series. The uniqueness criterion and stability of solutions with respect to the boundary data are established.