共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
In this paper, we present a general ellipsoidal artificial boundary method for three-dimensional exterior problem. The exact artificial boundary condition, which is expressed explicitly by the series concerning the ellipsoidal harmonic functions, is derived and then an equivalent problem in a bounded domain is presented. The error estimates show that the convergence rate depends on the mesh parameter, the number of terms used in the exact artificial boundary condition, and the location of the artificial boundary. 相似文献
3.
C. I. Goldstein 《Numerische Mathematik》1982,38(1):61-82
Summary The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL
2,H
1 andL
, provided that certain relationships hold between the frequency, mesh size and outer radius. 相似文献
4.
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
5.
Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation
of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded
computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family
of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local
ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend
on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary.
Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate
the performance of our error bounds.
Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002 相似文献
6.
Summary We provide a convergence rate analysis for a variant of the domain decomposition method introduced by Gropp and Keyes for solving the algebraic equations that arise from finite element discretization of nonsymmetric and indefinite elliptic problems with Dirichlet boundary conditions in 2. We show that the convergence rate of the preconditioned GMRES method is nearly optimal in the sense that the rate of convergence depends only logarithmically on the mesh size and the number of substructures, if the global coarse mesh is fine enough.This author was supported by the National Science Foundation under contract numbers DCR-8521451 and ECS-8957475, by the IBM Corporation, and by the 3M Company, while in residence at Yale UniversityThis author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under Contract W-31-109-Eng-38This author was supported by the National Science Foundation under contract number ECS-8957475, by the IBM Corporation, and by the 3M Company 相似文献
7.
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
8.
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
9.
Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit
layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce
accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified
SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it
yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the
severe nonuniformity of the mesh. We give local error estimates that hold true uniformly in the perturbation parameter , provided only that , where mesh points are used. Numerical experiments support these theoretical results.
Received February 19, 1999 / Revised version received January 27, 2000 / Published online August 2, 2000 相似文献
10.
Summary.
In an abstract framework we present a formalism which
specifies the notions of consistency and stability of
Petrov-Galerkin
methods used to approximate nonlinear problems which are, in many
practical situations, strongly nonlinear elliptic problems. This
formalism gives rise to a priori and a posteriori error estimates which
can be used for the refinement of the mesh in adaptive finite element
methods applied to elliptic nonlinear problems. This theory is
illustrated with the example: in a two
dimensional domain with Dirichlet boundary conditions.
Received June 10, 1992 / Revised version received February
28, 1994 相似文献
11.
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
12.
Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics
and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced
to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite
element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations
obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for
solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise
homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides
the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively
parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling
of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical
experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter.
Received August 28, 1996 / Revised version received March 10, 1997 相似文献
13.
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
14.
Xiaobo Liu 《Numerische Mathematik》1996,74(1):49-67
Summary. Interior error estimates are derived for a wide class of nonconforming finite element methods for second order scalar elliptic
boundary value problems. It is shown that the error in an interior domain can be estimated by three terms: the first one measures
the local approximability of the finite element space to the exact solution, the second one measures the degree of continuity
of the finite element space (the consistency error), and the last one expresses the global effect through the error in an
arbitrarily weak Sobolev norm over a slightly larger domain. As an application, interior superconvergences of some difference
quotients of the finite element solution are obtained for the derivatives of the exact solution when the mesh satisfies some
translation invariant condition.
Received December 29, 1994 相似文献
15.
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
16.
Zhongdi Cen Aimin Xu Anbo Le 《Journal of Computational and Applied Mathematics》2010,234(12):3445-3457
A system of coupled singularly perturbed initial value problems with two small parameters is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solution of the system has boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh a hybrid finite difference scheme is proved to be almost second-order accurate, uniformly in both small parameters. Numerical results supporting the theory are presented. 相似文献
17.
Summary. We consider the bilinear finite element approximation of smooth solutions to a simple parameter dependent elliptic model
problem, the problem of highly anisotropic heat conduction. We show that under favorable circumstances that depend on both
the finite element mesh and on the type of boundary conditions, the effect of parametric locking of the standard FEM can be
reduced by a simple variational crime. In our analysis we split the error in two orthogonal components, the approximation
error and the consistency error, and obtain different bounds for these separate components. Also some numerical results are
shown.
Received September 6, 1999 / Revised version received March 28, 2000 / Published online April 5, 2001 相似文献
18.
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
19.
Gerd Kunert 《Numerische Mathematik》2000,86(3):471-490
Summary. A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only.
The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence,
a so-called matching function is defined, and its discussion shows it to be a useful tool. With its help anisotropic interpolation estimates and subsequently
the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions in a posteriori error
analysis on isotropic and anisotropic meshes. A numerical example supports the anisotropic error analysis.
Received April 6, 1999 / Revised version received July 2, 1999 / Published online June 8, 2000 相似文献
20.
Summary. In this paper we consider two aspects of the problem of designing efficient numerical methods for the approximation of semilinear
boundary value problems. First we consider the use of two and multilevel algorithms for approximating the discrete solution.
Secondly we consider adaptive mesh refinement based on feedback information from coarse level approximations. The algorithms
are based on an a posteriori error estimate, where the error is estimated in terms of computable quantities only. The a posteriori
error estimate is used for choosing appropriate spaces in the multilevel algorithms, mesh refinements, as a stopping criterion
and finally it gives an estimate of the total error.
Received April 8, 1997 / Revised version received July 27, 1998 / Published online September 24, 1999 相似文献