Steady Stokes Problem Research Articles

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

AbstractWe study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the ‐setup. The mathematical model used here is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated using three types of boundary conditions: the conditions of periodicity on the “lower” and “upper” parts of the boundary, the Dirichlet boundary conditions on the “inflow” and on the profile and an artificial “do nothing”‐type boundary condition on the “outflow.” Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in and its continuous dependence on the data. We explain the sense in which the “do nothing” boundary condition on the “outflow” is satisfied.

A posteriori error analysis of an enriched Galerkin method of order one for the Stokes problem

Abstract We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ2, and tetrahedral or hexahedral meshes in ℝ3.

The Maximum Regularity Property of the Steady Stokes Problem Associated with a Flow Through a Profile Cascade

We deal with a steady Stokes-type problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain \(\varOmega \). The corresponding Stokes–type problem is formulated by means of the Stokes equation, equation of continuity and three types of boundary conditions: the conditions of periodicity on the curves \(\varGamma _{\hspace {-1.1pt} 0}\) and \(\varGamma _{\hspace {-1.1pt} 1}\), the Dirichlet boundary conditions on \(\varGamma _{\hspace {-1.1pt} \mathrm {in}}\) and \(\varGamma _{\hspace {-1.1pt} p}\) and an artificial “do nothing”–type boundary condition on \(\varGamma _{\hspace {-1.1pt} \mathrm {out}}\). (See Fig. 1.) We explain on the level of weak solutions the sense in which the last condition is satisfied. We show that, although domain \(\varOmega \) is not smooth and different types of boundary conditions meet in the corners of \(\varOmega \), the considered problem has a strong solution with the so called maximum regularity property.

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

SummaryMonolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared with preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this article, several techniques are applied to monolithic two‐level generalized Dryja‐Smith‐Widlund (GDSW) preconditioners to further improve the convergence speed and the computing time. In particular, reduced dimension GDSW coarse spaces, restricted and scaled versions of the first level, hybrid, and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time‐dependent Navier‐Stokes problem on 240 message passing interface (MPI) ranks, a reduction of 86% of the time‐to‐solution can be obtained. Even without applying recycling strategies, the time‐to‐solution can be reduced by more than 50% for a larger steady Stokes problem on 4608 MPI ranks. For the largest problems with 11 979 MPI ranks, the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved.

Pressure-induced locking in mixed methods for time-dependent (Navier–)Stokes equations

We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and Navier--Stokes equations, extending earlier work on the steady (Navier-)Stokes Problem. A locking phenomenon is identified for classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart elements by a novel, elegant and simple numerical analysis and corresponding numerical experiments, whenever the momentum balance is dominated by forces of a gradient type. More precisely, a reduction of the $L^2$ convergence order for high order methods, and even a complete stall of the $L^2$ convergence order for lowest-order methods on preasymptotic meshes is predicted by the analysis and practically observed. On the other hand, it is also shown that (structure-preserving) pressure-robust mixed methods do not suffer from this locking phenomenon, even if they are of lowest-order. A connection to well-balanced schemes for (vectorial) hyperbolic conservation laws like the shallow water or the compressible Euler equations is made.

A triangular spectral element method for elliptic and Stokes problems

In this paper, we study a triangular spectral-element method based on a one-to-one mapping between the rectangle and the triangle. We construct a new approximation space where the integral singularity brought by the mapping can be removed in a naive and stable way. We build aquasi-interpolation triangular spectral-element approximation, and analyze its approximation error. Based on this quasi-interpolation spectral-element approximation, we put forward a new triangular spectral-element method for the elliptic problems. We present the approximation scheme, analyze the convergence, and do some experiments to test the effectiveness. At last, we implement this triangular spectral-element method to solve the steady Stokes problem.

A compatible high-order meshless method for the Stokes equations with applications to suspension flows

A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.

On singular solutions of time-periodic and steady Stokes problems in a power cusp domain

The time-periodic and steady Stokes problems with the boundary value having a nonzero flux are considered in the power cusp domains. The asymptotic expansion near the singularity point is constructed in order to reduce the problem to the case where the energy solution exists. The solution of the problem is found then as the sum of the asymptotic expansion and the term with finite dissipation of energy.

A short note on L q theory for Stokes problem with a pressure-dependent viscosity

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on p and on the symmetric part of a gradient of u, namely, it is represented by a stress tensor T (Du, p):= v(p, |D|2)D which satisfies r-growth condition with r ∈ (1, 2]. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in the paper Caffarelli, Peral (1998).

Asymptotic conditions at infinity for the time-periodic Stokes problem in a system of pipes

A time-periodic Stokes problem set in a domain with cylindrical outlets to infinity is studied. We derive the generalized Green formula. It enables us to impose so-called asymptotic conditions at infinity which allow to determine a unique time-periodic solution. This method was proposed and applied for a steady Stokes problem by Nazarov and co-authors.

The NonConforming Virtual Element Method for the Stokes Equations

We present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable nonpolynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the nonconforming VEM is inf-sup stable and establish optimal a pri...

Regularity of solutions to the Navier-Stokes equations with a nonstandard boundary condition

In this paper we are concerned with the regularity of solutions to the Navier-Stokes equations with the condition on the pressure on parts of the boundary where there is flow. For the steady Stokes problem a result similar to Lq-theory for the one with Dirichlet boundary condition is obtained. Using the result, for the steady Navier-Stokes equations we obtain regularity as the case of Dirichlet boundary conditions. Furthermore, for the time-dependent 2-D Navier-Stokes equations we prove uniqueness and existence of regular solutions, which is similar to J.M.Bernard’s results[6] for the time-dependent 2-D Stokes equations.

A posteriorierror estimates of stabilized finite volume method for the Stokes equations

In this work, the residual‐type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity inH1norm and pressure inL2norm are derived. Furthermore, a global upper bound ofu−uhinL2‐norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.

On the Stokes and Oseen Problems with Singular Data

We consider the steady Stokes and Oseen problems in bounded and exterior domains of \({\mathbb{R}^n}\) of class Ck-1,1 (n = 2, 3; k ≥ 2). We prove existence and uniqueness of a very weak solution for boundary data a in \({W^{2-k-1/q,q} (\partial\Omega)}\) . If \({\Omega}\) is of class \({C^\infty}\) , we can assume a to be a distribution on \({\partial\Omega}\) .

Analysis of three stabilized finite volume iterative methods for the steady Navier–Stokes equations

In this work, three stabilized finite volume iterative schemes for the stationary Navier–Stokes equations are considered. Under the finite volume discretization at each iterative step, the iterative scheme I consists in solving the steady Stokes problem, iterative scheme II consists in solving the stationary linearized Navier–Stokes equations and iterative scheme III consists in solving the steady Oseen equations, respectively. We discuss the stabilities and convergence of three iterative methods. The iterative schemes I and II are stable and convergent under some strong uniqueness conditions, while iterative scheme III is unconditionally stable and convergent under the uniqueness condition. Finally, some numerical results are presented to verify the performance of these iterative schemes.

Prescription of General Defective Boundary Conditions in Fluid-Dynamics

This work reviews and extends to a more general setting some strategies to impose defective boundary conditions in fluid-dynamic problems investigated by the authors in the last years. We focus here to the steady Stokes problem. We show the well posedness of the proposed approaches and discuss their relative benefits.

FINITE VOLUME METHOD FOR GENERAL MULTIFLUID FLOWS GOVERNED BY THE INTERFACE STOKES PROBLEM

We study the approximation of solutions to the steady Stokes problem with a discontinuous viscosity coefficient (interface Stokes problem) in the 2D "Discrete Duality Finite Volume" (DDFV) framework. In order to take into account the discontinuities of the viscosity and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first show how to design our modified scheme, called m-DDFV, and we analyze its well-posedness and its convergence properties. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for Stokes problems.

A posteriori error estimations for mixed finite-element approximations to the Navier–Stokes equations

A posteriori estimates for mixed finite element discretizations of the Navier–Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier–Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.

The Discrete Duality Finite Volume Method for Stokes Equations on Three-Dimensional Polyhedral Meshes

We develop a discrete duality finite volume method for the three-dimensional steady Stokes problem with a variable viscosity coefficient on polyhedral meshes. Under very general assumptions on the mesh, which may admit nonconforming polyhedrons, we prove the stability and well-posedness of the scheme. We also prove the convergence of the numerical approximation to the velocity, velocity gradient, and pressure and derive a priori estimates for the corresponding approximation error. Final numerical experiments confirm the theoretical predictions.

Error Analysis for a Mimetic Discretization of the Steady Stokes Problem on Polyhedral Meshes

We present the development, convergence analysis, and numerical tests of the mimetic finite difference method for the Stokes problem on two-dimensional polygonal and three-dimensional polyhedral meshes.