共查询到20条相似文献,搜索用时 15 毫秒
1.
Multilevel Schwarz methods for elliptic problems
with discontinuous coefficients in three dimensions
Summary.
Multilevel Schwarz methods are developed for a
conforming finite element approximation of second order elliptic problems. We
focus on problems in three dimensions with
possibly large jumps in the coefficients across the
interface separating the subregions. We establish
a condition number estimate for the iterative operator, which is
independent of the coefficients, and grows at most as the square
of the number of levels. We also characterize a class of distributions
of the coefficients,
called quasi-monotone, for which the weighted
-projection is
stable and for which we can use the standard piecewise
linear functions as a coarse space. In this case,
we obtain optimal methods, i.e. bounds which are independent of the number
of levels and subregions. We also design and analyze multilevel
methods with new coarse spaces
given by simple explicit formulas. We consider nonuniform meshes
and conclude by an analysis of multilevel iterative substructuring methods.
Received April 6, 1994 / Revised version received December 7,
1994 相似文献
2.
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
3.
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
4.
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
5.
Finite element methods and their convergence for elliptic and parabolic interface problems 总被引:5,自引:0,他引:5
In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems
in two-dimensional convex polygonal domains. Nearly the same optimal -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but
are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation
are reasonable and practical.
Received July 7, 1996 / Revised version received March 3, 1997 相似文献
6.
The cascadic multigrid method for elliptic problems 总被引:23,自引:0,他引:23
Summary. The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods
based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade
principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed
error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on
finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, whichused the CG method as basic iteration method on each level. In [18] it has been proven,
that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations. The
present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid
smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly
highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and
robustness of the cascadic multigrid method.
Received November 12, 1994 / Revised version received October 12, 1995 相似文献
7.
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
8.
Summary. We consider a second-order elliptic equation with discontinuous or anisotropic coefficients in a bounded two- or three dimensional domain, and its finite-element discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients. Received February 5, 1999 / Published online March 16, 2000 相似文献
9.
Luca F. Pavarino 《Numerische Mathematik》1994,69(2):185-211
Summary.
In some applications, the accuracy of the numerical solution of an
elliptic problem needs to be increased only in certain parts of the
domain. In this paper, local refinement is introduced for an overlapping
additive Schwarz algorithm for the $-version finite element method.
Both uniform and variable degree refinements are considered.
The resulting algorithm is highly parallel and scalable.
In two and three dimensions,
we prove an optimal bound for the condition number of the iteration
operator under certain hypotheses on the refinement region.
This bound is independent of the degree $, the number of
subdomains $ and the mesh size $.
In the general two dimensional case, we prove an almost optimal bound
with polylogarithmic growth in $.
Received February 20, 1993 / Revised version received January
20, 1994 相似文献
10.
Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline
collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic
partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums
and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general
theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent
to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize
and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the
solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are
presented.
Received March 1, 1994 / Revised version received January 23, 1996 相似文献
11.
In this paper, we propose two variants of the additive Schwarz method for the approximation of second order elliptic boundary
value problems with discontinuous coefficients, on nonmatching grids using the lowest order Crouzeix-Raviart element for the
discretization in each subdomain. The overall discretization is based on the mortar technique for coupling nonmatching grids.
The convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements.
The condition number bound for the preconditioned systems is independent of the jumps of the coefficient, and depend linearly
on the ratio between the subdomain size and the mesh size. The performance of the methods is illustrated by some numerical
results.
This work has been supported by the Alexander von Humboldt Foundation and the special funds for major state basic research
projects (973) under 2005CB321701 and the National Science Foundation (NSF) of China (No.10471144)
This work has been supported in part by the Bergen Center for Computational Science, University of Bergen 相似文献
12.
Andrea Toselli 《Numerische Mathematik》2000,86(4):733-752
Summary. A two-level overlapping Schwarz method is considered for a Nédélec finite element approximation of 3D Maxwell's equations. For a fixed relative overlap, the condition number of the method is bounded, independently of the mesh size of the triangulation and the number of subregions. Our results are obtained with the assumption that the coarse triangulation is quasi-uniform and, for the Dirichlet problem, that the domain is convex. Our work generalizes well–known results for conforming finite elements for second order elliptic scalar equations. Numerical results for one and two-level algorithms are also presented. Received November 11, 1997 / Revised version received May 26, 1999 / Published online June 21, 2000 相似文献
13.
Least-squares mixed finite element methods
for non-selfadjoint elliptic problems: I. Error estimates
Summary.
A least-squares mixed finite element
method for general second-order non-selfadjoint
elliptic problems in two- and three-dimensional domains
is formulated and analyzed. The finite element spaces for
the primary solution approximation
and the flux approximation
consist of piecewise polynomials of degree
and respectively.
The method is mildly nonconforming on the boundary.
The cases and
are studied.
It is proved that the method is not subject to the LBB-condition.
Optimal - and
-error estimates are derived for
regular finite element partitions.
Numerical experiments, confirming the theoretical rates of
convergence, are presented.
Received
October 15, 1993 / Revised version received August 2, 1994 相似文献
14.
Summary. In this paper, we study a multiscale finite element method for solving a class of elliptic problems with finite number of
well separated scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving
all small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive
to the local property of the differential operator. The construction of the base functions is fully decoupled from element
to element; thus the method is perfectly parallel and is naturally adapted to massively parallel computers. We present the
convergence analysis of the method along with the results of our numerical experiments. Some generalizations of the multiscale
finite element method are also discussed.
Received April 17, 1998 / Revised version received March 25, 2000 / Published online June 7, 2001 相似文献
15.
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
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. In this paper we study the numerical behaviour of elliptic
problems in which a small parameter is involved and an example
concerning the computation of elastic arches is analyzed using this
mathematical framework. At first, the statements of the problem and its
Galerkin approximations are defined and an asymptotic
analysis is performed. Then we give general conditions ensuring that
a numerical scheme will converge uniformly with respect to the small
parameter. Finally we study an example in
computation of arches working in linear elasticity conditions. We build one
finite element scheme giving a locking behaviour, and another one
which does not.
Revised version received October 25, 1993 相似文献
18.
Summary. Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in
order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment
of non-homogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the
setting of space refinement (i.e. adaptive) discretizations of second order problems. Our method is based on the use of compatible
multiscale decompositions for both the domain and its boundary, and on the possibility of characterizing various function
spaces from the numerical properties of these decompositions. In particular, this allows the construction of a lifting operator
which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis.
An explicit construction of the wavelet bases and the lifting is proposed on fairly general domains, based on conforming domain decomposition techniques.
Received November 2, 1998 / Published online April 20, 2000 相似文献
19.
Finite volume element methods for non-definite problems 总被引:8,自引:0,他引:8
Ilya D. Mishev 《Numerische Mathematik》1999,83(1):161-175
Summary. The error estimates for finite volume element method applied to 2 and 3-D non-definite problems are derived. A simple upwind scheme is proven to be unconditionally stable and first order accurate. Received August 27, 1997 / Revised version received May 12, 1998 相似文献
20.
Ralf Kornhuber 《Numerische Mathematik》1996,72(4):481-499
Summary.
We derive globally convergent multigrid methods
for discrete elliptic
variational inequalities of the second kind
as obtained from
the approximation of related continuous
problems by piecewise linear finite elements.
The coarse grid corrections are computed
from certain obstacle problems.
The actual constraints are fixed by the
preceding nonlinear fine grid smoothing.
This new approach allows the implementation
as a classical V-cycle and preserves
the usual multigrid efficiency.
We give estimates
for the asymptotic convergence rates.
The numerical results indicate a significant improvement
as compared with previous multigrid approaches.
Received
March 26, 1994 / Revised version received September 22, 1994 相似文献