共查询到20条相似文献,搜索用时 390 毫秒
1.
Rob Stevenson 《Numerische Mathematik》1997,78(2):269-303
Summary. In this paper, we introduce a multi-level direct sum space decomposition of general, possibly locally refined linear or multi-linear
finite element spaces. The resulting additive Schwarz preconditioner is optimal for symmetric second order elliptic problems.
Moreover, it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh, perturbations with
positive zeroth order terms, and, after a further decomposition of the spaces, also with respect to anisotropy along the grid
lines. Important for an efficient implementation is that stable bases of the subspaces defining our decomposition, consisting
of functions having small supports can be easily constructed.
Received September 8, 1995 / Revised version received October 31, 1996 相似文献
2.
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
3.
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
4.
This paper is devoted to analysis of some convergent properties of both linear and quadratic simplicial finite volume methods
(FVMs) for elliptic equations. For linear FVM on domains in any dimensions, the inf-sup condition is established in a simple
fashion. It is also proved that the solution of a linear FVM is super-close to that of a relevant finite element method (FEM).
As a result, some a posterior error estimates and also algebraic solvers for FEM are extended to FVM. For quadratic FVM on
domains in two dimensions, the inf-sup condition is established under some weak condition on the grid. 相似文献
5.
The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H1-norm is proved to be O(h+H3|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method. 相似文献
6.
Daoqi Yang 《Numerische Mathematik》2002,93(1):177-200
Summary Iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations. The first one has
applications to elliptic equations and incompressible fluid flow problems, while the second has applications to linear elasticity
and compressible Stokes problems. These schemes are constructed through iteratively penalizing the mixed finite element scheme,
of which iterated penalty method and augmented Lagrangian method are special cases. Convergence theorems are demonstrated
in abstract formulations in Hilbert spaces, and applications to individual physical problems are considered as examples. Theoretical
analysis and computational experiments both show that the proposed schemes have very fast convergence; a few iterations are
normally enough to reduce the iterative error to a prescribed precision. Numerical examples with continuous and discontinuous
coefficients are presented. 相似文献
7.
Tarek P. Mathew 《Numerische Mathematik》1993,65(1):445-468
Summary We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by the Army Research Office under Grant DAAL 03-91-G-0150, while the author was a Visiting Assistant Researcher at UCLA 相似文献
8.
YunKyong Hyon Do Young Kwak 《Journal of Computational and Applied Mathematics》2011,235(14):4265-4271
In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory. 相似文献
9.
In the present paper we develop a representation of a norm frequently used in the analysis of multilevel methods. This allows
us to examine the convergence of additive Schwarz schemes also in the case of non-nested subspaces. We demonstrate the usefulness
of the given norm representation by studying in detail the stability of sparse grid splittings due to Griebel and Oswald,
which turns out to be a special case of our unified theory. Further applications concerning approximation spaces and non-nested
finite element spaces are given.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
10.
Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse
and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems
and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint
and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem
on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear
elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic
problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms.
A set of numerical examples are presented to confirm the estimates.
The work is supported by the National Natural Science Foundation of China (Grant No: 10601045). 相似文献
11.
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
12.
Olaf Steinbach 《Numerische Mathematik》2002,90(4):775-786
Summary. In this paper we investigate a stability estimate needed in hybrid finite and boundary element methods, especially in hybrid
coupled domain decomposition methods including mortar finite elements. This stability estimate is equivalent to the stability
of a generalized projection in certain Sobolev spaces. Using piecewise linear trial spaces and appropriate piecewise constant test spaces,
the stability of the generalized projection is proved assuming some mesh conditions locally.
Received April 11, 2000 / Revised version received February 15, 2001 / Published online July 25, 2001 相似文献
13.
Summary.
In this paper we introduce a class of robust multilevel
interface solvers for two-dimensional
finite element discrete elliptic problems with highly
varying coefficients corresponding to geometric decompositions by a
tensor product of strongly non-uniform meshes.
The global iterations convergence rate is shown to be of
the order
with respect to the number of degrees
of freedom on the single subdomain boundaries, uniformly upon the
coarse and fine mesh sizes, jumps in the coefficients
and aspect ratios of substructures.
As the first approach, we adapt the frequency filtering techniques
[28] to construct robust smoothers
on the highly non-uniform coarse grid. As an alternative, a multilevel
averaging procedure for successive coarse grid correction is
proposed and analyzed.
The resultant multilevel coarse grid
preconditioner is shown to have (in a two level case) the condition
number independent
of the coarse mesh grading and
jumps in the coefficients related to the coarsest refinement level.
The proposed technique exhibited high serial and parallel
performance in the skin diffusion processes modelling [20]
where the high dimensional coarse mesh problem inherits a strong geometrical
and coefficients anisotropy.
The approach may be also applied to magnetostatics problems
as well as in some composite materials simulation.
Received December 27, 1994 相似文献
14.
Nonlinear Galerkin methods and mixed finite elements:
two-grid algorithms for the Navier-Stokes equations 总被引:14,自引:0,他引:14
Summary.
A nonlinear Galerkin method using mixed finite
elements is presented for the two-dimensional
incompressible Navier-Stokes equations. The
scheme is based on two finite element spaces
and for the approximation of the velocity,
defined respectively on one coarse grid with grid
size and one fine grid with grid size and
one finite element space for the approximation
of the pressure. Nonlinearity and time
dependence are both treated on the coarse space.
We prove that the difference between the new
nonlinear Galerkin method and the standard
Galerkin solution is of the order of $H^2$, both in
velocity ( and pressure norm).
We also discuss a penalized version of our algorithm
which enjoys similar properties.
Received October 5, 1993 / Revised version received November
29, 1993 相似文献
15.
Summary.
We consider two level overlapping Schwarz domain decomposition methods
for solving the finite element problems that arise from
discretizations of elliptic problems on general unstructured meshes
in two and three dimensions. Standard finite element interpolation
from
the coarse to the fine grid may be used. Our theory requires no
assumption on the substructures
that constitute the whole domain, so the
substructures can be of arbitrary shape and of different
size. The global coarse mesh is allowed to be non-nested
to the fine grid on which the discrete problem is to be solved, and
neither
the coarse mesh nor the fine mesh need be quasi-uniform.
In addition, the domains defined by the fine and coarse grid need
not be identical. The one important constraint is that the closure
of the coarse grid must cover any portion of the fine grid boundary
for which Neumann boundary conditions are given.
In this general setting, our algorithms have the same optimal
convergence rate as the usual two level overlapping domain decomposition
methods on structured meshes.
The condition number of the preconditioned system depends only on the
(possibly small)
overlap of the
substructures and the size of the coarse grid, but is independent of
the sizes of the subdomains.
Received
March 23, 1994 / Revised version received June 2, 1995 相似文献
16.
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
17.
Summary For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable. 相似文献
18.
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
19.
Summary Subspace decompositions of finite element spaces based onL
2-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors have proved the equivalence of the associated discrete norms with theH
1-norm. The present paper gives an elementary, self-contained derivation of this result which is based on the use ofK-functionals known from the theory of interpolation spaces. 相似文献
20.
Domain decomposition for multiscale PDEs 总被引:3,自引:1,他引:2
We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic
PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved
by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous
media, in both the deterministic and (Monte–Carlo simulated) stochastic cases. We consider preconditioners which combine local
solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid.
We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on
the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this
preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse
grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions,
the preconditioner can still be robust even for large coefficient variation inside domains, when the classical method fails
to be robust. In particular our estimates prove very precisely the previously made empirical observation that the use of low-energy
coarse spaces can lead to robust preconditioners. We go on to consider coarse spaces constructed from multiscale finite elements
and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary (i.e.,
two-scale) media model problems. Moreover numerical experiments show that the new preconditioner has greatly improved performance
over standard preconditioners even in the random coefficient case. We show also how the analysis extends in a straightforward
way to multiplicative versions of the Schwarz method.
We would like to thank Bill McLean for very useful discussions concerning this work. We would also like to thank Maksymilian
Dryja for helping us to improve the result in Theorem 4.3. 相似文献