Incompressible Stokes Equations Research Articles

A new quadrilateral MINI-element for Stokes equations

We introduce a new stable MINI-element pair for incompressible Stokes equations on quadrilateral meshes, which uses the smallest number of bubbles for the velocity. The pressure is discretized with the P 1 -midpoint-edge-continuous elements and each component of the velocity field is done with the standard Q 1 -conforming elements enriched by one bubble a quadrilateral. The superconvergence in the pressure of the proposed pair is analyzed on uniform rectangular meshes, and tested numerically on uniform and non-uniform meshes.

The least-squares pseudo-spectral method for Navier–Stokes equations

A spectral collocation approximation of first-order system least squares for incompressible Stokes equations was analyzed in Kim et al. (2004) [12], and finite element approximations for incompressible Navier–Stokes equations were developed in Bochev et al. (1998,1999) [9,10]. The aim of this paper is to analyze the first-order system least-squares pseudo-spectral method for incompressible Navier–Stokes equations. The paper will be an extension of the result in Kim et al. (2004) [12] to the Navier–Stokes equations. Our least-squares functional is defined by the sum of discrete spectral norms of a first-order system of equations corresponding to the Navier–Stokes equations based on Legendre–Gauss–Lobatto points. We show its ellipticity and continuity over an appropriate product space, and spectral convergences of discretization errors are derived in the H1-norm and the L2-norm in each variable. Finally, we present some numerical examples.

A Nonoverlapping Domain Decomposition Method for Incompressible Stokes Equations with Continuous Pressures

A nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semidefinite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains fo...

A unified dual‐primal finite element tearing and interconnecting approach for incompressible Stokes equations

SUMMARYA unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd.

Sufficient conditions for second order L2‐convergence of the fractional step theta method

AbstractWe consider the fractional step theta time stepping procedure for the non‐stationary incompressible linear Stokes equations in a cylindrical domain (0, T) × Ω, where Ω is a bounded domain in ℝn. Using energy estimates and assuming a certain degree of regularity for the data, we show second order L2‐convergence. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multiscale Modeling of Colloid and Fluid Dynamics in Porous Media Including an Evolving Microstructure

We consider colloidal dynamics and single-phase fluid flow within a saturated porous medium in two space dimensions. A new approach in modeling pore clogging and porosity changes on the macroscopic scale is presented. Starting from the pore scale, transport of colloids is modeled by the Nernst–Planck equations. Here, interaction with the porous matrix due to (non-)DLVO forces is included as an additional transport mechanism. Fluid flow is described by incompressible Stokes equations with interaction energy as forcing term. Attachment and detachment processes are modeled by a surface reaction rate. The evolution of the underlying microstructure is captured by a level set function. The crucial point in completing this model is to set up appropriate boundary conditions on the evolving solid–liquid interface. Their derivation is based on mass conservation. As a result of an averaging procedure by periodic homogenization in a level set framework, on the macroscale we obtain Darcy’s law and a modified averaged convection–diffusion equation with effective coefficients due to the evolving microstructure. These equations are supplemented by microscopic cell problems. Time- and space-dependent averaged coefficient functions explicitly contain information of the underlying geometry and also information of the interaction potential. The theoretical results are complemented by numerical computations of the averaged coefficients and simulations of a heterogeneous multiscale scenario. Here, we consider a radially symmetric setting, i.e., in particular we assume a locally periodic geometry consisting of circular grains. We focus on the interplay between attachment and detachment reaction, colloidal interaction forces, and the evolving microstructure. Our model contributes to the understanding of the effects and processes leading to porosity changes and pore clogging from a theoretical point of view.

Analysis of a projection method for low-order nonconforming finite elements

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree non-conforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second-order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense and the optimal order of approximation in space is still achieved, even for open boundary conditions.

Time-Splitting Methods to Solve the Stochastic Incompressible Stokes Equation

For the stochastic incompressible time-dependent Stokes equation, we study different time-splitting methods that decouple the computation of velocity and pressure iterates in every iteration step. Optimal strong convergence is shown for Chorin's time-splitting scheme in the case of solenoidal noise, while computational counterexamples show poor convergence behavior in the case of general stochastic forcing. This suboptimal performance may be traced back to the nonregular pressure process in the case of general noise. A modified version of the deterministic time-splitting method that distinguishes between the deterministic and stochastic pressure removes this deficiency, leading to optimal convergence behavior.

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system $${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$$ in $${\mathbb {R}^3 \times (0,\infty)}$$ with initial data in Lebesgue spaces $${L^2(\mathbb {R}^3)}$$ or $${L^3(\mathbb {R}^3)}$$ . We analyze the convergence of its solutions to a solution of the incompressible Navier–Stokes system as $${\epsilon \to 0}$$ .

On iterative methods for the incompressible Stokes problem

AbstractIn this paper, we discuss various techniques for solving the system of linear equations that arise from the discretization of the incompressible Stokes equations by the finite‐element method. The proposed solution methods, based on a suitable approximation of the Schur‐complement matrix, are shown to be very effective for a variety of problems. In this paper, we discuss three types of iterative methods. Two of these approaches use the pressure mass matrix as preconditioner (or an approximation) to the Schur complement, whereas the third uses an approximation based on the ideas of least‐squares commutators (LSC). We observe that the approximation based on the pressure mass matrix gives h‐independent convergence, for both constant and variable viscosity. Copyright © 2010 John Wiley & Sons, Ltd.

A Finite Element Approximation of the Stokes Equations

In this work, a numerical solution of the incompressible Stokes equations is proposed. The method suggested is based on an algorithm of discretization by the unstable of Q1 – P0 velocity/pressure ?nite element approximation. It is shown that the inf-sup stability constant is O(h) in two dimensions and O( h2) in three dimensions. The basic tool in the analysis is the method of modi?ed equations which is applied to ?nite difference representations of the underlying ?nite element equations. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

A quadratic equal‐order stabilized method for Stokes problem based on two local Gauss integrations

AbstractIn this article, we analyze a quadratic equal‐order stabilized finite element approximation for the incompressible Stokes equations based on two local Gauss integrations. Our method only offsets the discrete pressure gradient space by the residual of the simple and symmetry term at element level to circumvent the inf‐sup condition. And this method does not require specification of a stabilization parameter, and always leads to a symmetric linear system. Furthermore, this method is unconditionally stable, and can be implemented at the element level with minimal additional cost. Finally, we give some numerical simulations to show good stability and accuracy properties of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

COUPLED CHEMOTAXIS FLUID MODEL

We consider a model system for the collective behavior of oxygen-driven swimming bacteria in an aquatic fluid. In certain parameter regimes, such suspensions of bacteria feature large-scale convection patterns as a result of the hydrodynamic interaction between bacteria. The presented model consist of a parabolic–parabolic chemotaxis system for the oxygen concentration and the bacteria density coupled to an incompressible Stokes equation for the fluid driven by a gravitational force of the heavier bacteria. We show local existence of weak solutions in a bounded domain in ℝd, d = 2, 3 with no-flux boundary condition and in ℝ2 in the case of inhomogeneous Dirichlet conditions for the oxygen.

The immersed boundary method for advection–electrodiffusion with implicit timestepping and local mesh refinement

We describe an immersed boundary method for problems of fluid–solute-structure interaction. The numerical scheme employs linearly implicit timestepping, allowing for the stable use of timesteps that are substantially larger than those permitted by an explicit method, and local mesh refinement, making it feasible to resolve the steep gradients associated with the space charge layers as well as the chemical potential, which is used in our formulation to control the permeability of the membrane to the (possibly charged) solute. Low Reynolds number fluid dynamics are described by the time-dependent incompressible Stokes equations, which are solved by a cell-centered approximate projection method. The dynamics of the chemical species are governed by the advection–electrodiffusion equations, and our semi-implicit treatment of these equations results in a linear system which we solve by GMRES preconditioned via a fast adaptive composite-grid (FAC) solver. Numerical examples demonstrate the capabilities of this methodology, as well as its convergence properties.

Buoyancy-driven convection around exothermic autocatalytic chemical fronts traveling horizontally in covered thin solution layers

Spatial variations of concentrations and temperature across exothermic chemical fronts can initiate buoyancy-driven convection. We investigate here theoretically the spatiotemporal dynamics arising from such a coupling between exothermic autocatalytic reactions, diffusion, and buoyancy-driven flows when an exothermic autocatalytic front travels perpendicularly to the gravity field in a thin solution layer. To do so, we numerically integrate the incompressible Stokes equations coupled to evolution equations for the concentration of the autocatalytic product and temperature through buoyancy terms proportional to, respectively, a solutal R(C) and a thermal R(T) Rayleigh number. We show that exothermic fronts can exhibit new types of dynamics in the presence of convection with regard to the isothermal system. In the cooperative case (R(C) and R(T) are of the same sign), the dynamics asymptotes to one vortex surrounding, deforming, and accelerating the front much like in the isothermal case. However, persistent local stratification of heavy zones over light ones can be observed at the rear of the front when the Lewis number Le (ratio of thermal diffusivity over molecular diffusion) is nonzero. When the solutal and thermal effects are antagonistic (R(C) and R(T) of opposite sign), temporal oscillations of the concentration, temperature, and velocity fields can, in some cases, be observed in a reference frame moving with the front. The various dynamical regimes are discussed as a function of R(C), R(T), and Le.

A nonlinear weighted least-squares finite element method for Stokes equations

The paper concerns a nonlinear weighted least-squares finite element method for the solutions of the incompressible Stokes equations based on the application of the least-squares minimization principle to an equivalent first order velocity–pressure–stress system. Model problem considered is the flow in a planar channel. The least-squares functional involves the L 2 -norms of the residuals of each equation multiplied by a nonlinear weighting function and mesh dependent weights. Using linear approximations for all variables, by properly adjusting the importance of the mass conservation equation and a carefully chosen nonlinear weighting function, the least-squares solutions exhibit optimal L 2 -norm error convergence in all unknowns. Numerical solutions of the flow pass through a 4 to 1 contraction channel will also be considered.

ANALYSIS AND COMPUTATIONS OF LEAST-SQUARES METHOD FOR OPTIMAL CONTROL PROBLEMS FOR THE STOKES EQUATIONS

First-order least-squares method of a distributed optimal con- trol problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equiv- alent first-order system by introducing the vorticity and then the least- squares functional corresponding to the system is defined in terms of the sum of the squared Hi1 and L2 norms of the residual equations of the system. Finite element approximations are studied and optimal error es- timates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational eciency.

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

The projection method is a widely used fractional-step algorithm for solving the incompressible Navier–Stokes equations. Despite numerous improvements to the methodology, however, imposing physical boundary conditions with projection-based fluid solvers remains difficult, and obtaining high-order accuracy may not be possible for some choices of boundary conditions. In this work, we present an unsplit, linearly-implicit discretization of the incompressible Navier–Stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations. Since our scheme is not a fractional-step algorithm, it is straightforward to specify general physical boundary conditions accurately; however, this capability comes at the price of having to solve the time-dependent incompressible Stokes equations at each timestep. To solve this linear system efficiently, we employ a Krylov subspace method preconditioned by the projection method. In our implementation, the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning. The accuracy of the scheme is demonstrated for several problems, including forced and unforced analytic test cases and lid-driven cavity flows. These tests consider a variety of physical boundary conditions with Reynolds numbers ranging from 1 to 30000. The effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the Schur complement for the time-dependent incompressible Stokes operator. The projection method is found to be a more efficient preconditioner in most cases considered in the present work.

An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries

We present an immersed interface method for solving the incompressible steady Stokes equations involving fixed/moving interfaces and rigid boundaries (irregular domains). The fixed/moving interfaces and rigid boundaries are represented by a number of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundaries, singular forces are applied on the fluid at these boundaries. The strength of singular forces at the rigid boundary is determined by solving a small system of equations. For the deformable interfaces, the forces that the interface exerts on the fluid are calculated from the configuration (position) of the deformed interface. The jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed/moving interfaces and rigid boundaries. These forces are interpolated using cubic splines and applied to the fluid through the jump conditions. The positions of the deformable interfaces are updated implicitly using a quasi-Newton method (BFGS) within each time step. In the proposed method, the Stokes equations are discretized via the finite difference method on a staggered Cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible Stokes flows with fixed/moving interfaces on irregular domains.

Buoyancy-driven convection around chemical fronts traveling in covered horizontal solution layers

Density differences across an autocatalytic chemical front traveling horizontally in covered thin layers of solution trigger hydrodynamic flows which can alter the concentration profile. We theoretically investigate the spatiotemporal evolution and asymptotic dynamics resulting from such an interplay between isothermal chemical reactions, diffusion, and buoyancy-driven convection. The studied model couples the reaction-diffusion-convection evolution equation for the concentration of an autocatalytic species to the incompressible Stokes equations ruling the evolution of the flow velocity in a two-dimensional geometry. The dimensionless parameter of the problem is a solutal Rayleigh number constructed upon the characteristic reaction-diffusion length scale. We show numerically that the asymptotic dynamics is one steady vortex surrounding, deforming, and accelerating the chemical front. This chemohydrodynamic structure propagating at a constant speed is quite different from the one obtained in the case of a pure hydrodynamic flow resulting from the contact between two solutions of different density or from the pure reaction-diffusion planar traveling front. The dynamics is symmetric with regard to the middle of the layer thickness for positive and negative Rayleigh numbers corresponding to products, respectively, lighter or heavier than the reactants. A parametric study shows that the intensity of the flow, the propagation speed, and the deformation of the front are increasing functions of the Rayleigh number and of the layer thickness. In particular, the asymptotic mixing length and reaction-diffusion-convection speed both scale as square root Ra for Ra>5. The velocity and concentration fields in the asymptotic dynamics are also found to exhibit self-similar properties with Ra. A comparison of the dynamics in the case of a monostable versus bistable kinetics is provided. Good agreement is obtained with experimental data on the speed of iodate-arsenous acid fronts propagating in horizontal capillaries. We furthermore compare the buoyancy-driven dynamics studied here to Marangoni-driven deformation of traveling chemical fronts in solution open to the air in the absence of gravity previously studied in the same geometry [L. Rongy and A. De Wit, J. Chem. Phys. 124, 164705 (2006)].