Stationary Stokes Problem Research Articles

An a posteriori error analysis for an augmented discontinuous Galerkin method applied to Stokes problem

AbstractThis paper deals with the a posteriori error analysis for an augmented mixed discontinuous formulation for the stationary Stokes problem. By considering an appropriate auxiliary problem, we derive an a posteriori error estimator. We prove that this estimator is reliable and locally efficient, and consists of just five residual terms. Numerical experiments confirm the theoretical properties of the augmented discontinuous scheme as well as of the estimator. They also show the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution.

The evolution Navier–Stokes equations in a cube under Navier boundary conditions: rarefaction and uniqueness of global solutions

We study the evolution Navier–Stokes equations in a cube under Navier boundary conditions. For the related stationary Stokes problem, we determine explicitly all the eigenvectors, eigenvalues and the corresponding Weyl asymptotic. We introduce the notion of rarefaction, namely families of eigenvectors that weakly interact with each other through the nonlinearity. By combining the spectral analysis with rarefaction, we expand the solutions in Fourier series, making explicit some of their properties. We then suggest several new points of view in order to explain the striking difference in uniqueness results between 2D and 3D. First, we construct examples of solutions for which the nonlinearity plays a minor role, both in 2D and 3D. Second, we show that, if a solution is rarefied, then its energy is decreasing: hence, rarefaction may be seen as an almost two dimensional assumption. Finally, by exploiting the explicit form of the eigenvectors we provide a numerical explanation of the difficulty in using energy methods for general solutions of the 3D equations.

Global well-posedness and convergence to equilibrium for the Abels-Garcke-Grün model with nonlocal free energy

We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existence of global strong solutions in general two-dimensional bounded domains and their uniqueness when the initial datum is strictly separated from the pure phases. The key points are a novel well-posedness result of strong solutions to the nonlocal convective Cahn-Hilliard equation with singular potential and constant mobility under minimal integral assumption on the incompressible velocity field, and a new two-dimensional interpolation estimate for the L4(Ω) control of the pressure in the stationary Stokes problem. Secondly, we show that any weak solution, whose existence was already known, is globally defined, enjoys the propagation of regularity and converges towards an equilibrium (i.e., a stationary solution) as t→∞. Furthermore, we demonstrate the uniqueness of strong solutions and their continuous dependence with respect to general (not necessarily separated) initial data in the case of matched densities and unmatched viscosities (i.e., the nonlocal model H with variable viscosity, singular potential and constant mobility). Finally, we provide a stability estimate between the strong solutions to the nonlocal AGG model and the nonlocal Model H in terms of the difference of densities.

Accelerating Algebraic Multigrid Methods via Artificial Neural Networks

We present a novel deep learning-based algorithm to accelerate—through the use of Artificial Neural Networks (ANNs)—the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations stemming from finite element discretizations of Partial Differential Equations (PDE). We show that ANNs can be successfully used to predict the strong connection parameter that enters in the construction of the sequence of increasingly smaller matrix problems standing at the basis of the AMG algorithm, so as to maximize the corresponding convergence factor of the AMG scheme. To demonstrate the practical capabilities of the proposed algorithm, which we call AMG-ANN, we consider the iterative solution of the algebraic system of equations stemming from finite element discretizations of two-dimensional model problems. First, we consider an elliptic equation with a highly heterogeneous diffusion coefficient and then a stationary Stokes problem. We train (off-line) our ANN with a rich dataset and present an in-depth analysis of the effects of tuning the strong threshold parameter on the convergence factor of the resulting AMG iterative scheme.

Regularity estimates for stationary Stokes problem in some generalized function spaces

Regularity estimates for stationary Stokes problem in some generalized function spaces

Inf-sup stability implies quasi-orthogonality

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms and leads to simple optimality proofs for the Taylor-Hood discretization of the stationary Stokes problem, a finite-element/boundary-element discretization of an unbounded transmission problem, and an adaptive time-stepping scheme for parabolic equations. The main technical tools are new stability bounds for the L U LU -factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

A locally stabilized collocated finite volume method for the stationary Stokes problem

In this paper, a novel discrete formulation for stabilizing the collocated finite volume method (CFVM) is introduced and analyzed for the stationary Stokes problem. This method is based on the lowest equal‐order approximation (piecewise constants) for both velocity and pressure unknowns. A macro‐volume (i.e., a small group of volumes) condition is introduced to build the local stabilized finite volume formulation for the problem. We prove that a stabilization term based on the jump of the discrete pressures over the interior edges of the macro‐volume gives a stable scheme. Moreover, the existence and uniqueness of the discrete solution are proved. The error estimate of the stabilized finite volume formulation is derived. Finally, some numerical tests are performed to exhibit the stability and effectiveness of the proposed method.

Nonconforming P1 Finite Element for the Biharmonic Problem and Its Application to Stokes Problem

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming [Formula: see text] finite element. For the solutions which admit a regularity greater than [Formula: see text], in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in [Formula: see text] an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem.

Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems

<p style='text-indent:20px;'>We consider a boundary value problem for the stationary Stokes problem and the corresponding pressure-Poisson equation. We propose a new formulation for the pressure-Poisson problem with an appropriate additional boundary condition. We establish error estimates between solutions to the Stokes problem and the pressure-Poisson problem in terms of the additional boundary condition. As boundary conditions for the Stokes problem, we use a traction boundary condition and a pressure boundary condition introduced in C. Conca et al (1994).

A posteriori estimates for the stationary Stokes problem in exterior domains

A posteriori estimates for the stationary Stokes problem in exterior domains

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

We study the interrelation between the limit $$L_p(\Omega )$$-Sobolev regularity $$\overline{s}_p$$ of (classes of) functions on bounded Lipschitz domains $$\Omega \subseteq \mathbb {R}^d$$, $$d\ge 2$$, and the limit regularity $$\overline{\alpha }_p$$ within the corresponding adaptivity scale of Besov spaces $$B^\alpha _{\tau ,\tau }(\Omega )$$, where $$1/\tau =\alpha /d+1/p$$ and $$\alpha >0$$ ($$p>1$$ fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N-term approximation. We show how additional information on the Besov or Triebel–Lizorkin regularity may be used to deduce upper bounds for $$\overline{\alpha }_p$$ in terms of $$\overline{s}_p$$ simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton et al. (in: De Carli and Milman (ed) Interpolation theory and applications, American Mathematical Society, Providence, 2007). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

New Stability Estimates for an Unfitted Finite Element Method for Two-Phase Stokes Problem

The paper addresses stability and finite element analysis of the stationary two-phase Stokes problem with a piecewise constant viscosity coefficient experiencing a jump across the interface between...

Estimates of the Distance to Exact Solutions of the Stokes Problem with Slip and Leak Boundary Conditions

We deduce a posteriori error estimates of functional type for the stationary Stokes problem with slip and leak boundary conditions. The derived error majorants do not contain mesh dependent constants and are valid for a wide class of energy admissible approximations that satisfy the Dirichlet boundary condition on a part of the boundary. Different forms of error majorants contain global constants associated with Poincare type inequalities or the stability (LBB) condition for the Stokes problem or constants associated with subdomains (if a domain decomposition is applied). It is proved that the majorants are guaranteed and vanish if and only if the functions entering them coincide with the respective exact solutions.

Optimality of a Standard Adaptive Finite Element Method for the Stokes Problem

We prove that the a standard adaptive algorithm for the Taylor--Hood discretization of the stationary Stokes problem converges with optimal rate. This is done by developing an abstract framework for quite general problems, which allows us to prove general quasi-orthogonality proposed in [C. Carstensen et al., Comput. Math. Appl., 67 (2014), pp. 1195--1253]. This property is the main obstacle towards the optimality proof and therefore is the main focus of this work. The key ingredient is a new connection between the mentioned quasi-orthogonality and $LU$-factorization of infinite matrices.

Regularity for elliptic systems of differential forms and applications

We prove existence and up to the boundary regularity estimates in $L^{p}$ and H\"{o}lder spaces for weak solutions of the linear system $$ \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text{ in } \Omega, $$ with either $ \nu\wedge \omega$ and $\nu\wedge \delta \left( B\omega \right)$ or $\nu\lrcorner B\omega$ and $\nu\lrcorner \left( A d\omega \right)$ prescribed on $\partial\Omega.$ The proofs are in the spirit of `Campanato method' and thus avoid potential theory and do not require a verification of Agmon-Douglis-Nirenberg or Lopatinski\u{i}-Shapiro type conditions. Applications to a number of related problems, such as general versions of the time-harmonic Maxwell system, stationary Stokes problem and the `div-curl' systems, are included.

Interior discontinuity for a stationary compressible Stokes system with inflow datum

In this paper we show an interior discontinuity and its piecewise regularity for a stationary compressible Stokes problem with inflow jump datum. The discontinuity is preserved along the interior curve directed by the ambient vector field. The difficulty lies in controlling the gradient of the pressure across the interior curve starting at the jump point that the inflow datum has. We deal with this issue by constructing a vector function K corresponding to the interior jump and splitting the vector K from the velocity vector. Finally the remainder for the velocity vector and pressure function is shown to be in the space H2,q×H1,q for q≥2.

An error analysis of the finite element method overcoming corner singularities for the stationary Stokes problem

In this paper, we introduce a mixed finite element method to overcome corner singularities of the stationary Stokes system on a non-convex polygon. The reference Choi and Kweon (2013) says that the velocity field and the pressure function are composed of singular parts and regular remainders near a non-convex vertex, and furthermore the regular parts are sufficiently smooth on the polygon. We use the corner singularity expansion and propose new extraction formulae for coefficients of singularities. For the proposed numerical method, we first try to find finite element solutions for the regular parts and then calculate approximations of the coefficients by the derived formulae expressed by the remainders and given functions. We show error estimates of the approximations for the regular parts and coefficients, and give some numerical experiments to confirm the efficiency and reliability of the proposed method.

Analysis of a Stokes interface problem in multi-subdomains

This paper studies a stationary Stokes problem with piecewise constant viscosities in multi-subdomains. For the variational formulation of this problem, we establish its validity of inf–sup stability with the constant depending on the second smallest viscosity.

A least-squares finite element method for a nonlinear Stokes problem in glaciology

A stationary Stokes problem with nonlinear rheology and with mixed no-slip and sliding basal boundary conditions is considered. The model describes the flow of ice in glaciers and ice sheets. A least-squares finite element method is developed and analyzed. The method does not require that the finite element spaces satisfy an inf–sup condition. Moreover, the usage of negative Sobolev norm in the least-squares functional allows for the use of standard piecewise polynomials spaces for both the velocity and pressure approximations. A Picard-type iterative method is used to linearize the Stokes problem. It is shown that the linearized least-squares functional is coercive and continuous in an appropriate solution space so the existence and uniqueness of a weak solution immediately follows as do optimal error estimates for finite element approximations. Numerical tests are provided to illustrate the theory.

Justification of lubrication approximation: An application to fluid/solid interactions

We consider the stationary Stokes problem in a three-dimensional fluid domain $\mathcal F$ with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle $\mathcal B$ in a bounded or an exterior smooth container $\Omega$. We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small.