共查询到20条相似文献,搜索用时 15 毫秒
1.
2.
Summary.
Discretisation of the classical Stokes problem gives rise
to symmetric indefinite matrices with eigenvalues which,
in a precise way, are not symmetric about the origin, but which
do depend on a mesh size parameter. Convergence
estimates for the Conjugate Residual or Minimum Residual
iterative solution of such systems are given by best
minimax polynomial approximations on an inclusion set for the
eigenvalues.
In this paper, an analytic convergence estimate for such
problems is given in terms of an asymptotically small
mesh size parameter.
Received
November 16, 1993 / Revised version received August 2,
1994 相似文献
3.
Susanne C. Brenner 《Numerische Mathematik》1999,83(2):187-203
Summary. It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order
, where d measures the diameters of the subdomains and h is the mesh size of the triangulation. The result holds for both conforming and nonconforming finite elements.
Received: January 15, 1998 相似文献
4.
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
5.
Summary. An unusual stabilized finite element is presented and analyzed herein for a generalized Stokes problem with a dominating
zeroth order term. The method consists in subtracting a mesh dependent term from the formulation without compromising consistency.
The design of this mesh dependent term, as well as the stabilization parameter involved, are suggested by bubble condensation.
Stability is proven for any combination of velocity and pressure spaces, under the hypotheses of continuity for the pressure
space. Optimal order error estimates are derived for the velocity and the pressure, using the standard norms for these unknowns.
Numerical experiments confirming these theoretical results, and comparisons with previous methods are presented.
Received April 26, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001
Correspondence to: Gabriel R. Barrenechea 相似文献
6.
Klaus Wilderotter 《Numerische Mathematik》1997,75(3):397-404
Summary. Let be the unit disk in the complex plane and let be a compact, simply connected subset of , whose boundary is assumed to belong to the class . Let be the unit ball of the Hardy space . A linear algorithm is constructed for approximating functions in . The algorithm is based on sampling functions in the Fejer points of and it produces the error Here denotes the space of continuous functions on and is the Green capacity of with respect to . Moreover it is shown that the algorithm is asymptotically optimal in the sense of -widths.
Received July 7, 1994 相似文献
7.
Jae Ryong Kweon 《Numerische Mathematik》2000,86(2):305-318
Summary. We formulate the compressible Stokes system given in (1.1) into a (new) weak formulation (2.1). A finite element method for
this is presented. Existence and uniqueness of the finite element method is shown. An optimal error estimate for the numerical
approximation is obtained. Numerical examples are given, showing its efficiency and rates of convergence of the approximate
solutions that results from the discrete problem (3.1).
Received October 20, 1996 / Revised version received January 21, 1999 / Published online: April 20, 2000 相似文献
8.
Summary.
We develop and analyze a procedure for creating a hierarchical basis of
continuous piecewise linear polynomials on an arbitrary, unstructured,
nonuniform triangular
mesh. Using these hierarchical basis functions, we are able to define
and analyze corresponding iterative methods for solving the linear
systems arising from finite element discretizations of elliptic
partial differential equations. We show that such iterative methods
perform as well as those developed for the usual case of
structured, locally refined meshes. In particular, we show that the
generalized condition numbers for such iterative methods are
of order ,
where is the number of hierarchical basis levels.
Received December 5, 1994 相似文献
9.
Summary. A unified approach to construct finite elements based on a dual-hybrid formulation of the linear elasticity problem is given.
In this formulation the stress tensor is considered but its symmetry is relaxed by a Lagrange multiplier which is nothing
else than the rotation. This construction is linked to the approximations of the Stokes problem in the primitive variables
and it leads to a new interpretation of known elements and to new finite elements. Moreover all estimates are valid uniformly
with respect to compressibility and apply in the incompressible case which is close to the Stokes problem.
Received June 20, 1994 / Revised version received February 16, 1996 相似文献
10.
Summary. We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity. For this problem, the
classical finite element method of degree one converges only in for the norm of the vorticity. We propose to use harmonic functions to approach the vorticity along the boundary. Discrete harmonics
are functions that are used in practice to derive a new numerical method. We prove that we obtain with this numerical scheme
an error of order for the norm of the vorticity.
Received January, 2000 / Revised version received May 15, 2001 / Published online December 18, 2001 相似文献
11.
Summary.
We consider the mixed formulation for the
elasticity problem and the limiting
Stokes problem in ,
.
We derive a set of sufficient conditions under which families of
mixed finite element spaces
are simultaneously stable with respect to the mesh size
and, subject to a
maximum loss of
,
with respect to the polynomial
degree .
We obtain asymptotic
rates of convergence that are optimal up to
in the
displacement/velocity and up to
in the
"pressure", with
arbitrary
(both rates being
optimal with respect to
). Several choices of
elements are discussed with reference to
properties desirable in the
context of the -version.
Received
March 4, 1994 / Revised version received February 12, 1995 相似文献
12.
Summary. The aim of this paper is to give a new method for the numerical approximation of the biharmonic problem. This method is based
on the mixed method given by Ciarlet-Raviart and have the same numerical properties of the Glowinski-Pironneau method. The
error estimate associated to these methods are of order O(h) for k The algorithm proposed in this paper converges even for k, without any regularity condition on or . We have an error estimate of order O(h) in case of regularity.
Received February 5, 1999 / Revised version received February 23, 2000 / Published online May 4, 2001 相似文献
13.
In this paper, a multigrid algorithm is presented for the mortar element method for P1 nonconforming element. Based on the
theory developed by Bramble, Pasciak, Xu in [5], we prove that the W-cycle multigrid is optimal, i.e. the convergence rate
is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid preconditioner is constructed, which
results in a preconditioned system with uniformly bounded condition number.
Received May 11, 1999 / Revised version received April 1, 2000 / Published online October 16, 2000 相似文献
14.
To the best knowledge of the authors, this work presents the first convergence analysis for the Infinite Element Method (IEM)
for the Helmholtz equation in exterior domains. The approximation applies to separable geometries only, combining an arbitrary
Finite Element (FE) discretization on the boundary of the domain with a spectral-like approximation in the “radial” direction,
with shape functions resulting from the separation of variables. The principal idea of the presented analysis is based on
the spectral decomposition of the problem.
Received February 10, 1996 / Revised version received February 17, 1997 相似文献
15.
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
16.
Summary. The dimensional reduction method for solving boundary value problems of Helmholtz's equation in domain by replacing them with systems of equations in dimensional space are investigated. It is proved that the existence and uniqueness for the exact solution and the dimensionally reduced solution of the boundary value problem if the input data on the faces are in some class of functions. In addition, the difference
between and in is estimated as and are fixed. Finally, some numerical experiments in a domain are given in order to compare theretical results.
Received April 2, 1996 / Revised version received July 30, 1990 相似文献
17.
Oliver G. Ernst 《Numerische Mathematik》1996,75(2):175-204
Summary. We introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the Helmholtz equation.
The problem is reformulated as an equivalent one on a bounded domain using an exact non-local boundary condition on a circular
artificial boundary. An FFT-based fast Helmholtz solver is then derived for a finite-element discretization on an annular
domain. The exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method. The
imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to
mesh independent convergence rates when Krylov subspace methods are used to solve the capacitance matrix equation.
Received May 2, 1995 相似文献
18.
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
19.
This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the
outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision
quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions
are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space,
and that the Peano constant of the rule is sufficiently small. It is then shown that the semi-discrete Galerkin approximation is well posed. For the
trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous
piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis
functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete
Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the and norms.
Received March 15, 1996 / Revised version received June 2, 1997 相似文献
20.
W. Sun 《Numerische Mathematik》1995,70(4):501-506
Summary.
A direct method is developed for solving Poisson's equation on
an annulus
region with Hermite bicubic collocation approximation. In terms
of FFT, the
operation count is
on an mesh.
Received
May 11, 1990 / Revised version received August 19, 1994 相似文献