共查询到20条相似文献，搜索用时 15 毫秒
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We develop multilevel augmentation methods for solving differential equations. We first establish a theoretical framework
for convergence analysis of the boundary value problems of differential equations, and then construct multiscale orthonormal
bases in H0m(0,1) spaces. Finally, the multilevel augmentation methods in conjunction with the multiscale orthonormal bases are applied
to two-point boundary value problems of both second-order and fourth-order differential equations. Theoretical analysis and
numerical tests show that these methods are computationally stable, efficient and accurate.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem.
Mathematics subject classifications (2000) 65J15, 65R20.
Zhongying Chen: Supported in part by the Natural Science Foundation of China under grants 10371137 and 10201034, the Foundation
of Doctoral Program of National Higher Education of China under grant 20030558008, Guangdong Provincial Natural Science Foundation
of China under grant 1011170 and the Foundation of Zhongshan University Advanced Research Center.
Yuesheng Xu: Corresponding author. Supported in part by the US National Science Foundation under grants 9973427 and 0312113,
by NASA under grant NCC5-399, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of
Sciences under the program of “One Hundred Distinguished Young Scientists”. 相似文献
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Janne Martikainen Tuomo Rossi Jari Toivanen 《Numerical Linear Algebra with Applications》2002,9(8):629-652
A fast direct solution method for a discretized vector‐valued elliptic partial differential equation with a divergence constraint is considered. Such problems are typical in many disciplines such as fluid dynamics, elasticity and electromagnetics. The method requires the problem to be posed in a rectangle and boundary conditions to be either periodic boundary conditions or the so‐called slip boundary conditions in one co‐ordinate direction. The arising saddle‐point matrix has a separable form when bilinear finite elements are used in the discretization. Based on a result for so‐called p‐circulant matrices, the saddle‐point matrix can be transformed into a block‐diagonal form by fast Fourier transformations. Thus, the fast direct solver has the same structure as methods for scalar‐valued problems which are based on Fourier analysis and, therefore, it has the same computational cost ??(N log N). Numerical experiments demonstrate the good efficiency and accuracy of the proposed method. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
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Lukas Kogler Philip L. Lederer Joachim Schöberl 《Numerical Linear Algebra with Applications》2023,30(5):e2503
We are studying the efficient solution of the system of linear equations stemming from the mass conserving stress-yielding (MCS) discretization of the Stokes equations. We perform static condensation to arrive at a system for the pressure and velocity unknowns. An auxiliary space preconditioner for the positive definite velocity block makes use of efficient and scalable solvers for conforming Finite Element spaces of low order and is analyzed with emphasis placed on robustness in the polynomial degree of the discretization. Numerical experiments demonstrate the potential of this approach and the efficiency of the implementation. 相似文献
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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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Yueqiang Shang 《Numerical Methods for Partial Differential Equations》2013,29(6):2025-2046
A finite element variational multiscale method based on two local Gauss integrations is applied to solve numerically the time‐dependent incompressible Navier–Stokes equations. A significant feature of the method is that the definition of the stabilization term is derived via two local Guass integrations at element level, making it more efficient than the usual projection‐based variational multiscale methods. It is computationally cheap and gives an accurate approximation to the quantities sought. Based on backward Euler and Crank–Nicolson schemes for temporal discretization, we derive error bounds of the fully discrete solution which are first and second order in time, respectively. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
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We study numerical methods for a mixed Stokes/Darcy model in porous media applications. The global model is composed of two different submodels in a fluid region and a porous media region, coupled through a set of interface conditions. The weak formulation of the coupled model is of a saddle point type. The mixed finite element discretization applied to the saddle point problem leads to a coupled, indefinite, and nonsymmetric linear system of algebraic equations. We apply the preconditioned GMRES method to solve the discrete system and are particularly interested in efficient and effective decoupled preconditioning techniques. Several decoupled preconditioners are proposed. Theoretical analysis and numerical experiments show the effectiveness and efficiency of the preconditioners. Effects of physical parameters on the convergence performance are also investigated. 相似文献
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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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The paper presents finite element error estimates of a variational multiscale method (VMS) for the incompressible Navier–Stokes equations. The constants in these estimates do not depend on the Reynolds number but on a reduced Reynolds number or on the mesh size of a coarse mesh. This work is partially supported by NSF grants DMS9972622, DMS20207627 and INT9814115. 相似文献
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This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N8) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to Nd+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
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Yueqiang Shang 《Numerical Methods for Partial Differential Equations》2015,31(3):856-875
Based on fully overlapping domain decomposition and a recent variational multiscale method, a parallel finite element variational multiscale method for convection dominated incompressible flows is proposed and analyzed. In this method, each processor computes a local finite element solution in its own subdomain using a global mesh that is locally refined around its own subdomain, where a stabilization term based on two local Gauss integrations is adopted to stabilize the numerical form of the Navier–Stokes equations. Using the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 856–875, 2015 相似文献
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何成 《数学物理学报(B辑英文版)》1997,17(4)
1IntroductionConsidertheequationwhereaisaconstantin(0,1).ThesolutionisclearlyWenowomittheviscositytermandaboundaryconditionof(1.1)asfollows:Thesolutionof(1.2)isForesmallandx>E,thetwosolutionsareclosetogether.Theregionwheretheyaredrasticallydifferentisconfinedtotheinterval[0,E],whichisthesthcalledboundarylayer.Notesthatase~0,theboundarylayershrinkstozero,hiltthemaximumdifferencebetweenIReceivedMar.9,1995;revisedSep.18,1996.ThisworkissupportedbyFoundationoflhstituteofMathematics,Acadedasin… 相似文献
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An evolution compressible Stokes system is studied in a bounded cylindrical region . The initial datum of pressure is assumed to have a jump at a specified curve C0 in Ω. As predicted by the Rankine-Hugoniot conditions, the pressure and velocity derivatives have jump discontinuities along the characteristic plane of the curve C0 directed by an ambient velocity vector. An explicit formula for the jump discontinuity is presented. The jump decays exponentially in time, more rapidly for smaller viscosities. Under suitable conditions of the data, a regularity of the solution is established in a compact subregion of Q away from the jump plane. 相似文献
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Helmut Abels 《Journal of Evolution Equations》2002,2(4):439-457
In this article we prove the existence of bounded purely imaginary powers of the Stokes operator , which is defined on the space of solenoidal vector fields < q < , where is an infinite layer. It is a consequence of a special representation of the resolvent of the Stokes operator in terms of
the Stokes operator on , a composition of a trace and a Poisson operator – a singular Green operator – and a negligible part. 相似文献
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O. Goubet 《Applied Mathematics and Optimization》1996,34(3):361-365
In this paper we establish that the pressure gradient and the flux, for a linear stationary Stokes problem for general periodic two-dimensional channels, are related by a simple formula, the same as that for laminar Poiseuille flows. 相似文献
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Min Huang 《Advances in Computational Mathematics》2006,25(1-3):7-22
In this paper, a construction of multiscale bases for Petrov–Galerkin methods for Fredholm integral equations of the second
kind is proposed. The properties of multiscale bases are presented including additional order of vanishing moments, compact
supports and stability.
Communicated by A. Zhou Dedicated to Professor Charles A. Micchelli on the occasion of his sixtieth birthday with friendship
and esteem
Mathematics subject classifications (2000) 41A10, 65R20, 65D15.
Min Huang: Supported in part by Professor Yuesheng Xu's support under the program of “One Hundred Distinguished Young Scientists”
of the Chinese Academy of Sciences and by the Graduate Innovation Foundation of the Chinese Academy of Sciences. 相似文献
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We propose a new consistent, residual-based stabilization of the Stokes problem. The stabilizing term involves a pseudo-differential operator, defined via a wavelet expansion of the test pressures. This yields control on the full -norm of the resulting approximate pressure independently of any discretization parameter. The method is particularly well suited for being applied within an adaptive discretization strategy. We detail the realization of the stabilizing term through biorthogonal spline wavelets, and we provide some numerical results.
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《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(7):1829-1868
We prove the homogenisation to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. Due to the incompressibility condition and the lack of a maximum principle for the Stokes equations, our proof requires a very careful study of the geometry of the random holes generated by the class of probability measures considered. 相似文献
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We present a Lagrangian–Eulerian strategy for proving uniqueness and local existence of solutions of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations. 相似文献