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1.
In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions. 相似文献
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Jae Ryong Kweon 《Numerical Methods for Partial Differential Equations》2004,20(3):412-431
A linearized compressible viscous Stokes system is considered. The a posteriori error estimates are defined and compared with the true error. They are shown to be globally upper and locally lower bounds for the true error of the finite element solution. Some numerical examples are given, showing an efficiency of the estimator. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 412–431, 2004. 相似文献
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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 in H1 norm and pressure in L2 norm are derived. Furthermore, a global upper bound of u ? uh in L2‐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. 相似文献
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Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains 下载免费PDF全文
In this paper we consider the system of the non‐steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator and formulate our problem in terms of operator equations. Let and be the Fréchet derivative of at . We prove that is one‐to‐one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W2, 2‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data. 相似文献
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F. Guilln-Gonzlez M. A. Rodríguez-Bellido M. A. Rojas-Medar 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2004,21(6):783-826
The main subject of this work is to study the concept of very weak solution for the hydrostatic Stokes system with mixed boundary conditions (non-smooth Neumann conditions on the rigid surface and homogeneous Dirichlet conditions elsewhere on the boundary). In the Stokes framework, this concept has been studied by Conca [Rev. Mat. Apl. 10 (1989)] imposing non-smooth Dirichlet boundary conditions.In this paper, we introduce the dual problem that turns out to be a hydrostatic Stokes system with non-free divergence condition. First, we obtain strong regularity for this dual problem (which can be viewed as a generalisation of the regularity results for the hydrostatic Stokes system with free divergence condition obtained by Ziane [Appl. Anal. 58 (1995)]). Afterwards, we prove existence and uniqueness of very weak solution for the (primal) problem.As a consequence of this result, the existence of strong solution for the non-stationary hydrostatic Navier-Stokes equations is proved, weakening the hypothesis over the time derivative of the wind stress tensor imposed by Guillén-González, Masmoudi and Rodríguez-Bellido [Differential Integral Equations 50 (2001)]. 相似文献
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Vladimir Jovanovi Christian Rohde 《Numerical Methods for Partial Differential Equations》2005,21(1):104-131
We consider a class of finite‐volume schemes on unstructured meshes for symmetric hyperbolic linear systems of balance laws in two and three space dimensions. This class of schemes has been introduced and analyzed by Vila and Villedieu ( 5 ). They have proven an a priori error estimate for approximations of smooth solutions. We extend the results to weak solutions. This is the base to derive an a posteriori error estimate for finite‐volume approximations of weak solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
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Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions 下载免费PDF全文
Hailong Qiu Liquan Mei Yongchao Zhang 《Numerical Methods for Partial Differential Equations》2017,33(2):546-569
Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017 相似文献
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In this paper, we propose a spectral method for the vorticity‐stream function form of the Navier–Stokes equations with slip boundary conditions. The numerical solutions fulfill the incompressibility and the physical boundary conditions automatically. The stability and convergence of the proposed methods are proven. Numeric results demonstrate the efficiency of suggested algorithm. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
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In this paper, we consider the axisymmetric Navier-Stokes equations, and provide a refined a priori estimate for the swirl component of the vorticity. This extends Theorem 2 of [D. Chae, J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645--671]. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(5):2119-2139
In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P1−P1 or P1−P0) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results. 相似文献
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Sergey Korotov 《Applications of Mathematics》2007,52(3):235-249
The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is
strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in
the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model
consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered
in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper
and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation
measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used
to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true
error as the resources of the used computer allow.
This work was supported by the Academy Research Fellowship No. 208628 from the Academy of Finland. 相似文献
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《Numerical Methods for Partial Differential Equations》2018,34(2):401-418
In this article, we consider the finite element discretization of the Navier‐Stokes problem coupled with convection‐diffusion equations where both the viscosity and the diffusion coefficients depend on the temperature. Existence and uniqueness of a solution are established. We prove a posteriori error estimates. 相似文献
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Mark Steinhauer Jos Miguel Urbano Juha Videman 《Mathematical Methods in the Applied Sciences》2006,29(11):1339-1348
This note bridges the gap between the existence and regularity classes for the third‐grade Rivlin–Ericksen fluid equations. We obtain a new global a priori estimate, which conveys the precise regularity conditions that lead to the existence of a global in time regular solution. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
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Carlo Lovadina Mikko Lyly Rolf Stenberg 《Numerical Methods for Partial Differential Equations》2009,25(1):244-257
We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and lower a‐posteriori error bounds. The estimates are verified by numerical computations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
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We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR2, with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle. 相似文献
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Roland Becker Daniela Capatina Julie Joie 《Numerical Methods for Partial Differential Equations》2012,28(3):1013-1041
We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ?1 to the solution of some stable and well‐known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM‐solution based on the reconstruction of a locally conservative H(div, Ω)‐tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM‐solution. These results can be used to affirm the robustness of the DGFEM‐method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011 相似文献
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Ben‐yu Guo 《Mathematical Methods in the Applied Sciences》2008,31(5):607-626
In this paper, we consider incompressible viscous fluid flows with slip boundary conditions. We first prove the existence of solutions of the unsteady Navier–Stokes equations in n‐spacial dimensions. Then, we investigate the stability, uniqueness and regularity of solutions in two and three spacial dimensions. In the compactness argument, we construct a special basis fulfilling the incompressibility exactly, which leads to an efficient and convergent spectral method. In particular, we avoid the main difficulty for ensuring the incompressibility of numerical solutions, which occurs in other numerical algorithms. We also derive the vorticity‐stream function form with exact boundary conditions, and establish some results on the existence, stability and uniqueness of its solutions. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献