Two-dimensional Stokes Problem Research Articles

Conforming virtual element approximations of the two-dimensional Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes. The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the L2 and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.

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.

A remark on the non-uniqueness in L^infty of the solutions to the two-dimensional Stokes problem in exterior domains

The paper is concerned with the IBVP in exterior domains of the two-dimensional Stokes equations. The goal was to investigate the well-posedness in the set of solutions assuming an initial data u_0in L^infty (Omega ), divergence free, and enjoying the property for all t>0 and c independent of u. For all u_0in L^infty , divergence-free one shows examples of non-uniqueness in the above set of solutions.

The MINI mixed finite element for the Stokes problem: An experimental investigation

Oh3∕2 superconvergence in pressure and velocity has been experimentally investigated for the two-dimensional Stokes problem discretized with the MINI mixed finite element. Even though the classic mixed finite element theory for the MINI element guarantees linear convergence for the total error, recent theoretical results indicate that superconvergence of order Oh3∕2 in pressure and of the linear part of the computed velocity to the piecewise-linear nodal interpolation of the exact velocity is in fact possible with structured, three-directional triangular meshes. The numerical experiments presented here suggest a more general validity of Oh3∕2 superconvergence, possibly to automatically generated and unstructured triangulations. In addition, the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise-linear part of the computed velocity, finding that the former is generally closer to the exact velocity, whereas the latter conserves mass better.

Non-Darcy effects in fracture flows of a yield stress fluid

We study non-inertial flows of single-phase yield stress fluids along uneven/rough-walled channels, e.g. approximating a fracture, with two main objectives. First, we re-examine the usual approaches to providing a (nonlinear) Darcy-type flow law and show that significant errors arise due to self-selection of the flowing region/fouling of the walls. This is a new type of non-Darcy effect not previously explored in depth. Second, we study the details of flow as the limiting pressure gradient is approached, deriving approximate expressions for the limiting pressure gradient valid over a range of different geometries. Our approach is computational, solving the two-dimensional Stokes problem along the fracture, then upscaling. The computations also reveal interesting features of the flow for more complex fracture geometries, providing hints about how to extend Darcy-type approaches effectively.

FETI-DP preconditioners for a staggered discontinuous Galerkin formulation of the two-dimensional Stokes problem

In this paper, a class of FETI-DP preconditioners is developed for a fast solution of the linear system arising from staggered discontinuous Galerkin discretization of the two-dimensional Stokes equations. The discretization has been recently developed and has the distinctive advantages that it is optimally convergent and has a good local conservation property. In order to efficiently solve the linear system, two kinds of FETI-DP preconditioners, namely, lumped and Dirichlet preconditioners, are considered and analyzed. Scalable bounds C(H/h) and C(1+log(H/h))2 are proved for the lumped and Dirichlet preconditioners, respectively, with the constant C depending on the inf–sup constant of the discrete spaces but independent of any mesh parameters. Here H/h stands for the number of elements across each subdomain. Numerical results are presented to confirm the theoretical estimates.

Meshless boundary node methods for Stokes problems

In this study, we present the first meshless boundary node method (BNM) for two-dimensional and three-dimensional Stokes problems. The BNM exploits the advantages of the reduced dimensionality of boundary integral equations (BIEs) and the meshless attribute of moving least squares (MLS) approximations. However, because MLS shape functions lack the property of a delta function, the direct collocation scheme used in the BNM to impose boundary conditions doubles the numbers of unknowns and system equations. To overcome this drawback, we propose a dual boundary node method (DBNM), which uses the velocity BIE on the velocity boundary and the traction BIE on the traction boundary. In the DBNM, the boundary conditions are incorporated directly into the BIEs and they can be imposed easily, while the numbers of unknowns and system equations are only half of those in the BNM, thereby leading to higher computational precision and speed. Selected numerical tests illustrate the efficiency of the BNM and the DBNM.

Solving inverse Stokes problems by modified collocation Trefftz method

In this paper, the two-dimensional inverse Stokes problems, governed by bi-harmonic equations, are stably solved by the modified collocation Trefftz method (MCTM). In some practical applications of the Stokes problems, part of the boundary conditions cannot be measured in advance, so the mathematical descriptions of such problems are known as the inverse Stokes problems. When numerical simulation is adopted for solutions of the inverse Stokes problems, the solutions will become extremely unstable, which means that small perturbations in the boundary conditions will result in large errors of the final results. Hence, we adopted the MCTM for stably and efficiently analyzing the inverse Stokes problems. The MCTM is one kind of boundary-type meshless methods, so the mesh generation and the numerical quadrature can be avoided. Besides, the numerical solution is expressed as a linear combination of T-complete functions modified by a characteristic length. By enforcing the satisfactions of the boundary conditions at every boundary node, a system of linear algebraic equations will be yielded. The unknown coefficients in the solution expression can be acquired by directly inverting the coefficient matrix. The numerical solutions and their derivatives can be easily obtained by linear summation. Three numerical examples are provided to demonstrate the accuracy and the stability of the proposed meshless scheme for solving the two-dimensional inverse Stokes problems.

Solving the inverse Stokes problems by the modified collocation Trefftz method and Laplacian decomposition

In this paper, a boundary-type meshfree algorithm is proposed to accurately and stably deal with the two-dimensional inverse Stokes problems, which are highly ill-conditioned. Based on the Laplacian decomposition, the Stokes equations are recast as three Laplace equations. Then the modified collocation Trefftz method (MCTM), one of the most promising boundary-type meshless methods, is adopted to solve these three Laplace equations. The MCTM can stabilize the numerical scheme and obtain highly accurate results by utilizing the characteristic length. Accordingly, the numerical solutions of these three Laplace equations are expressed by linear combination of the modified T-complete functions. The unknown coefficients in the solution expressions are found by enforcing the satisfactions of the boundary conditions at the boundary collocation points. Three numerical examples are provided to show the efficacy and stability of the proposed meshless method. Besides, noises are added into the boundary conditions to demonstrate the stability of the proposed scheme for dealing with the inverse Stokes problems.

Stokes Complexes and the Construction of Stable Finite Elements with Pointwise Mass Conservation

Two families of conforming finite elements for the two-dimensional Stokes problem are developed, guided by two discrete smoothed de Rham complexes, which we coin “Stokes complexes.” We show that the finite element pairs are inf-sup stable and also provide pointwise mass conservation on very general triangular meshes.

A two-level nonoverlapping Schwarz algorithm for the Stokes problem: Numerical study

A general framework of a two-level nonoverlapping Schwarz algorithm for the Stokes problem is developed by relaxing average zero condition on pressure unknowns. This framework allows both discontinuous and continuous pressure finite element spaces. The coarse problem is built by algebraic manipulation after selecting appropriate primal unknowns just like in BDDC algorithms. Performance of the suggested algorithm is presented depending on the selection of finite elements and primal unknowns. Under the same set of primal unknowns, the algorithm for the case with discontinuous pressure functions outperforms one with continuous pressure functions. For the two-dimensional Stokes problem, the algorithm with a set of primal unknowns consisting of velocity unknowns at corners, averages of velocity components over common edges, and pressure unknowns at corners presents good scalability when continuous pressure test functions are used. In both two- and three-dimensional Stokes problems, an improvement can be made for the case with continuous pressure test functions by applying the suggested algorithm to the interface problem, which is obtained by eliminating velocity unknowns and pressure unknowns interior to each subdomains.

THE MODIFIED COLLOCATION TREFFTZ METHOD AND LAPLACIAN DECOMPOSITION FOR SOLVING TWO-DIMENSIONAL STOKES PROBLEMS

In this paper, the two-dimensional Stokes problem is analyzed by the modified collocation Trefftz method (MCTM) and the Laplacian decomposition. The coupled Stokes equations are converted to three Laplace equations by utilizing the Laplacian decomposition and then the boundary-type meshless MCTM is adopted to solve the resultant Laplace equations. The MCTM, free from mesh and numerical quadrature, is derived from the conventional Trefftz method by considering the characteristic length of the domain, which stabilize the numerical scheme and obtain highly accurate results. Besides, the solutions are expressed as the linear combination of T-complete functions and the velocity as well as pressure are coupled by collocating the boundary conditions. Several numerical examples are provided to demonstrate the efficacy and accuracy of the proposed meshless scheme. In addition, the numerical results demonstrates that the proposed meshless scheme can solve the Stokes problems accurately in simplyand doubly-connected domains.

Penalty-Factor-Free Discontinuous Galerkin Methods for 2-Dim Stokes Problems

This paper presents two new finite element methods for two-dimensional Stokes problems. These methods are developed by relaxing the constraints of the Crouzeix-Raviart nonconforming $P_1$ finite elements. Penalty terms are introduced to compensate for lack of continuity or the divergence-free property. However, there is no need for choosing penalty factors, and the formulations are symmetric. These new methods are easy to implement and avoid solving saddle-point linear systems. Numerical experiments are presented to illustrate the proved optimal error estimates.

A Multigrid Method for the Pseudostress Formulation of Stokes Problems

The purpose of this paper is to develop and analyze a multigrid solver for the finite element discretization of the pseudostress system associated with the differential operator $\mathcal{A}-\gamma\,\mathbf{grad}\,\mathbf{div}$ over $2\times 2$ matrix-valued functions. This system is derived from the pseudostress-velocity formulation [Z. Cai, B. Lee, and P. Wang, SIAM J. Numer. Anal., 42 (2004), pp. 843-859] of two-dimensional Stokes problems through the penalty method or natural time discretization for the respective steady- or unsteady-state problems. Here $\gamma>0$ is a constant associated with either the penalty parameter or the time-step size, and $\mathcal{A}$ is a singular map. In this paper, we develop a multigrid solver for the discrete problem using both the Raviart-Thomas (RT) and the Brezzi-Douglas-Marini (BDM) finite elements. We show that the multigrid convergence rate is $O(\frac{1+\gamma^{-2}}{1+\gamma^{-2} + m})$, where $m$ is the number of smoothings. This convergence rate is independent of the mesh size and the number of levels used in multigrid. Moreover, numerical results presented in this paper show that the multigrid convergence rate for the BDM elements does not depend on the parameter $\gamma$.

A Neumann--Dirichlet Preconditioner for a FETI-DP Formulation of the Two-Dimensional Stokes Problem with Mortar Methods

A FETI-DP (dual-primal finite element tearing and interconnecting) formulation for the two-dimensional Stokes problem with mortar methods is considered. Separate sets of unknowns are used for velocity on interfaces, and the mortar constraints are enforced on the velocity unknowns by Lagrange multipliers. Average constraints on edges are further introduced as primal constraints to solvethe Stokes problem correctly and to obtain a scalable FETI-DP algorithm. A Neumann--Dirichlet preconditioner is shown to give a condition number bound, C maxi = 1,...,N {(1 + log( Hi / hi ))2}, where Hi and hi are the subdomain size and the mesh size, respectively, and the constant C is independent of the mesh parameters Hi and hi .

Vorticity-velocity-pressure formulation for Stokes problem

We propose a three-field formulation for efficiently solving a two-dimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.

New Implementation Techniques for the Exterior Stokes Problem in the Plane

We present a method to solve numerically two-dimensional Stokes problems on exterior domains. Our scheme is based on the fully discrete BEM–FEM formulation proposed in [21] whose main advantage is that only elemental quadratures are used to approximate the weakly singular boundary integrals. We show in this article that it is possible to maintain this important property without using curved triangles in the discretization process. This modification makes the method easier to implement and the numerical experiments reveal that it still keeps the optimal order of convergence of the original scheme.We also introduce in this paper a new iterative method to solve the complicated linear systems of equations that arises from this type of BEM–FEM discretizations.

SOLVING THE STOKES PROBLEM ON A MASSIVELY PARALLEL COMPUTER

We describe a numerical procedure for solving the stationary two-dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method.

A Fully Discrete BEM-FEM for the Exterior Stokes Problem in the Plane

We reformulate the Johnson--Nedelec approach for the exterior two-dimensional Stokes problem taking advantage of the parameterization of the artificial boundary. The main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem. This one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals. We give a complete error analysis of both the Galerkin and fully discrete Galerkin methods.

An optimal multilevel preconditioner for solenoidal approximations of the two-dimensional Stokes problem

We develop an optimal multilevel preconditioner for the divergence-free part of a modified conforming P1-P0 discretization of the two-dimensional Stokes problem which contains a novel prolongation operator preserving the discrete divergence-free property. The proofs utilize the equivalence to a discretization of the biharmonic Dirichlet problem by Powell-Sabin macrotriangles via a discrete streamfunction. It is shown how the basic preconditioner can be applied to other Stokes discretizations with discontinuous pressure elements.