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1.
Pengzhan Huang 《Applications of Mathematics》2014,59(4):361-370
This paper presents a superconvergence result based on projection method for stabilized finite element approximation of the Stokes eigenvalue problem. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The paper complements the work of Li et al. (2012), which establishes the superconvergence result of the Stokes equations by the stabilized finite element method. Moreover, numerical tests confirm the theoretical analysis. 相似文献
2.
In this paper, we present the preconditioned nullspace method for the iterative solution of the three-dimensional Stokes problem. In the nullspace method, the original saddle point system is reduced to a positive definite problem by representing the solution with respect to a basis of discretely divergence free vectors. The exact, explicit computation of such a basis typically has non-optimal (storage and computational) complexity. There exist some algorithms that exploit the sparsity of the matrix and work well for two dimensional problems but fail for three dimensions. Here, we will exploit an implicit representation of the nullspace basis which can be computed efficiently also in a three-dimensional setting, possibly only as an approximation. We will present some numerical results to illustrate the performance of the resulting solution method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
3.
We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization of the penalty term, we propose reduced and non-reduced integration schemes, and obtain an error estimate for velocity and pressure. The theoretical results are verified by numerical experiments. 相似文献
4.
We represent a new numerical method to solve the stationary Navier–Stokes equations in an unbounded domain. This technique consists in coupling the boundary integral and finite element methods. Moreover, we derive the variational formulation and well-posedness of the coupling method and provide the convergence result for the approximate solution. © 1993 John Wiley & Sons, Inc. 相似文献
5.
In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method
is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P
1–P
1 pair). After a relationship between this method and a stabilized finite element method is established, an error estimate
of optimal order in the H
1-norm for velocity and an estimate in the L
2-norm for pressure are obtained. An optimal error estimate in the L
2-norm for the velocity is derived under an additional assumption on the body force.
This work is supported in part by the NSF of China 10701001 and by the US National Science Foundation grant DMS-0609995 and
CMG Chair Funds in Reservoir Simulation. 相似文献
6.
Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces 总被引:1,自引:0,他引:1
Katrin Schumacher 《Annali dell'Universita di Ferrara》2008,54(1):123-144
We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt
weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier–Stokes equations
with integrable low-regularity data. Int. Math. Ser., pp. 1–26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi
et al. (Math. Ann. 331, 41–74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to
more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according
to such data.
相似文献
7.
Summary We propose a multidomain spectral collocation scheme for the approximation of the two-dimensional Stokes problem. We show that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.Deceased 相似文献
8.
V. A. Solonnikov 《Journal of Mathematical Sciences》1998,92(6):4364-4385
The main result is an asymptotic formula for a solution to the conjugation problem for the Navier-Stokes equations describing
the slow motion of two immiscible liquids such that one of them occupies a bounded domain Ω1 ⊂ ℝ3, whereas the other occupies the exterior domain Ω2=ℝ4∖Ω. Such a formula was obtained for a solution to the exterior problem with sticking conditions on the boundary in the works
of Fischer, Hsiao, and Wendland. The result obtained is applied to the proof of the solvability of a free-boundary problem
describing a uniform drop in an infinite liquid. Bibliography: 10 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 208–238. 相似文献
9.
Emmanuel Creus Serge Nicaise 《Numerical Methods for Partial Differential Equations》2006,22(2):449-483
This article presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e., elements with very large aspect ratio. Our analysis covers two‐ and three‐dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh, which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
10.
11.
We propose a new mixed formulation of the Stokes problem where the extra stress tensor is considered. Based on such a formulation, a mixed finite element is constructed and analyzed. This new finite element has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass. Optimal error estimates are derived. For the numerical implementation of this finite element, a hybrid form is presented. This work is a first step towards the treatment of viscoelastic fluid flows by mixed finite element methods. 相似文献
12.
We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By extending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results. 相似文献
13.
In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence
of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between
the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing
projection method. In particular, the stabilization of the continuous lowest equal order pair finite volume element discretization
is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries.
Finally, some numerical experiments that confirm the predicted behavior of the method are provided. 相似文献
14.
15.
We prove the unique solvability of solutions in Sobolev spaces to the stationary Stokes system on a bounded Reifenberg flat domain when the coefficients are partially BMO functions, i.e., locally they are merely measurable in one direction and have small mean oscillations in the other directions. Using this result, we establish the unique solvability in Muckenhoupt type weighted Sobolev spaces for the system with partially BMO coefficients on a Reifenberg flat domain. We also present weighted a priori -estimates for the system when the domain is the whole Euclidean space or a half space. 相似文献
16.
G. Matthies P. Skrzypacz L. Tobiska 《Numerical Methods for Partial Differential Equations》2005,21(4):701-725
For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q2 ? P element applied to the 3D stationary Stokes and Navier‐Stokes problem, respectively. Moreover, applying a Q3 ? P postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q2‐interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost‐benefit analysis between the two third‐order methods, the post‐processed Q2 ? P discretization, and the Q3 ? P discretization is carried out. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
17.
给出一种求解线性矩问题的逼近方法,并给出以B样条函数为基的数值例子,证明了该方法的有效性. 相似文献
18.
Summary.
Discretisation of the classical Stokes problem gives rise
to symmetric indefinite matrices with eigenvalues which,
in a precise way, are not symmetric about the origin, but which
do depend on a mesh size parameter. Convergence
estimates for the Conjugate Residual or Minimum Residual
iterative solution of such systems are given by best
minimax polynomial approximations on an inclusion set for the
eigenvalues.
In this paper, an analytic convergence estimate for such
problems is given in terms of an asymptotically small
mesh size parameter.
Received
November 16, 1993 / Revised version received August 2,
1994 相似文献
19.
Summary We study in this paper a new mixed finite element approximation of the Stokes' problem in the velocity pressure formulation. This approximation which is based on a new variational principle allows the use of low order Lagrange elements and leads to optimal order of convergence for the velocity and the pressure. Iterative and direct methods for the solution of the approximate problems will be discussed in a forthcoming paper. 相似文献