共查询到20条相似文献,搜索用时 15 毫秒
1.
E. F. Kaasschieter 《BIT Numerical Mathematics》1989,29(4):824-849
Discretizing a symmetric elliptic boundary value problem by a finite element method results in a system of linear equations with a symmetric positive definite coefficient matrix. This system can be solved iteratively by a preconditioned conjugate gradient method. In this paper a preconditioning matrix is proposed that can be constructed for all finite element methods if a mild condition for the node numbering is fulfilled. Such a numbering can be constructed using a variant of the reverse Cuthill-McKee algorithm. 相似文献
2.
Susanne C. Brenner 《Numerische Mathematik》1996,72(4):419-447
Summary.
A two-level additive Schwarz preconditioner is
developed for the
systems resulting from the discretizations of
the plate bending problem by the Morley finite element, the
Fraeijs de Veubeke finite element, the Zienkiewicz finite
element and the Adini
finite element. The condition numbers of the preconditioned
systems are shown
to be bounded independent of mesh sizes and the number of
subdomains in the
case of a generous overlap.
Received
February 1, 1994 / Revised version received October 24, 1994 相似文献
3.
Zhiming Chen 《Numerische Mathematik》2001,88(4):641-659
Summary. The Signorini problem describes the contact of a linearly elastic body with a rigid frictionless foundation. It is transformed
into a saddle point problem of some augmented Lagrangian functional and then discretized by finite element methods. Optimal
error estimates are obtained for general smooth domains which are not necessarily convex. The key ingredient in the analysis
is a discrete inf-sup condition which guarantees the existence of the saddle point.
Received January 29, 1999 / Revised version received May 2, 2000 / Published online December 19, 2000 相似文献
4.
Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the
exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm
by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples
show the optimal order of convergence.
Received July 8, 1997 /Revised version received October 23, 1997 相似文献
5.
I. Perugia 《Numerische Mathematik》1999,84(2):305-326
Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and
analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship,
while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic
field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements,
and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches
for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A
finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates.
Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999 相似文献
6.
Crouzeix-Raviart type finite elements on anisotropic meshes 总被引:47,自引:0,他引:47
Summary. The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element
is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is
proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from
classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity
assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate,
namely the Poisson problem in domains with edges. A numerical test is described.
Received May 19, 1999 / Revised version received February 2, 2000 / Published online February 5, 2001 相似文献
7.
A preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed. The proposed preconditioner
is of block triangular form. The aim of this article is to show that a rigorous convergence analysis can be performed by using
the field of values of the preconditioned linear system. As an example, a saddle point problem obtained from a mixed finite
element discretization of the Oseen equations is considered. The convergence estimates obtained by using a field–of–values
analysis are independent of the discretization parameter h. Several computational experiments supplement the theoretical results and illustrate the performance of the method.
Received March 20, 1997 / Revised version received January 14, 1998 相似文献
8.
A cascadic multigrid algorithm for semilinear elliptic problems 总被引:12,自引:0,他引:12
Gisela Timmermann 《Numerische Mathematik》2000,86(4):717-731
Summary. We propose a cascadic multigrid algorithm for a semilinear elliptic problem. The nonlinear equations arising from linear
finite element discretizations are solved by Newton's method. Given an approximate solution on the coarsest grid on each finer
grid we perform exactly one Newton step taking the approximate solution from the previous grid as initial guess. The Newton
systems are solved iteratively by an appropriate smoothing method. We prove that the algorithm yields an approximate solution
within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that
the initial grid size is sufficiently small. Moreover, we show that the method has multigrid complexity.
Received February 12, 1998 / Revised version received July 22, 1999 / Published online June 8, 2000 相似文献
9.
Jan H. Brandts Sergey Korotov Michal K?í?ek 《Linear algebra and its applications》2008,429(10):2344-2357
This paper provides a sufficient condition for the discrete maximum principle for a fully discrete linear simplicial finite element discretization of a reaction-diffusion problem to hold. It explicitly bounds the dihedral angles and heights of simplices in the finite element partition in terms of the magnitude of the reaction coefficient and the spatial dimension. As a result, it can be computed how small the acute simplices should be for the discrete maximum principle to be valid. Numerical experiments suggest that the bound, which considerably improves a similar bound in [P.G. Ciarlet, P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Eng. 2 (1973) 17-31.], is in fact sharp. 相似文献
10.
Carsten Carstensen 《Numerische Mathematik》1999,82(4):577-597
Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step
in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and
a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity.
The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear
resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows
linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite
element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This
estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded
from above. The constants depend on the hardening involved and become larger for decreasing hardening.
Received May 7, 1997 / Revised version received August 31, 1998 相似文献
11.
Olaf Steinbach 《Numerische Mathematik》2001,88(2):367-379
Summary. In this paper we prove the stability of the projection onto the finite element trial space of piecewise polynomial, in particular, piecewise linear basis functions in
for . We formulate explicit and computable local mesh conditions to be satisfied which depend on the Sobolev index s. In conclusion we prove a stability condition needed in the numerical analysis of mixed and hybrid boundary element methods
as well as in the construction of efficient preconditioners in adaptive boundary and finite element methods.
Received October 14, 1999 / Revised version received March 24, 2000 / Published online October 16, 2000 相似文献
12.
Anisotropic mesh refinement
in stabilized Galerkin methods 总被引:8,自引:0,他引:8
Summary.
The numerical solution of a convection-diffusion-reaction model problem is
considered in two and three dimensions. A stabilized finite element method
of Galerkin/Least-square type accomodates diffusion-dominated as well as
convection- and/or reaction-dominated situations. The resolution of
boundary layers occuring in the singularly perturbed case is achieved
using anisotropic mesh refinement in boundary layer regions.
In this paper, the
standard analysis of the stabilized Galerkin method on isotropic meshes
is extended to more general meshes with boundary layer refinement.
Simplicial Lagrangian elements of arbitrary order are used.
Received
March 6, 1995 / Revised version received August 18,
1995 相似文献
13.
Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes
Summary. We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite
bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional,
order m on a bounded (open or closed) surface of dimension d, with . We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively
subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially
with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts
to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown
to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio.
In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite
element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions.
Received October 5, 2001 / Revised version received December 5, 2001 / Published online April 17, 2002
The support of this work through Visiting Fellowship grant GR/N21970 from the Engineering and Physical Sciences Research
Council of Great Britain is gratefully acknowledged. The second author was also supported by the Australian Research Council 相似文献
14.
In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their
discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete
flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting
the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas
vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence
free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite
element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient
spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof.
Received November 4, 1996 / Revised version received February 2, 1998 相似文献
15.
Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems.This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter.In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter =1.The 2
nd
author's research was sponsored by Control Data Corporation through its PACER fellowship program.The 3
rd
author's research was supported by the Netherlands organization for scientific research (NWO).On leave from the Institute of Mathematics, Academy of Science, 1090 Sofia, P.O. Box 373, Bulgaria. 相似文献
16.
Rodolfo Araya Gabriel R. Barrenechea Abner Poza 《Journal of Computational and Applied Mathematics》2008
In this work we present an adaptive strategy (based on an a posteriori error estimator) for a stabilized finite element method for the Stokes problem, with and without a reaction term. The hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. An equivalence result between the norm of the finite element error and the estimator is given, where the dependence of the constants on the physics of the problem is explicited. Several numerical results confirming both the theoretical results and the good performance of the estimator are given. 相似文献
17.
Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and
analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency
components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid
by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained
in this paper for the finite element solutions on general shape-regular grids.
Y. He was partially subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703;
A. Zhou was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic
Research Program under the grant 2005CB321704; J. Li was partially supported by the NSF of China under the grant 10701001.
J. Xu was partially supported by Alexander von Humboldt Research Award for Senior US Scientists, NSF DMS-0609727 and NSFC-10528102. 相似文献
18.
Mark Ainsworth 《Journal of Computational and Applied Mathematics》2010,234(9):2618-2632
We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’. 相似文献
19.
Rob Stevenson 《Numerische Mathematik》2002,91(2):351-387
Summary. We derive sufficient conditions under which the cascadic multi-grid method applied to nonconforming finite element discretizations
yields an optimal solver. Key ingredients are optimal error estimates of such discretizations, which we therefore study in
detail. We derive a new, efficient modified Morley finite element method. Optimal cascadic multi-grid methods are obtained
for problems of second, and using a new smoother, of fourth order as well as for the Stokes problem.
Received February 12, 1998 / Revised version received January 9, 2001 / Published online September 19, 2001 相似文献
20.
Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized
Stokes problem (the Stokes problem with the additional term in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution
of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence
is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent
of . Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are
provided.
Received May 2, 1997 / Revised version received May 10, 1999 / Published online May 8, 2000 相似文献