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1.
《Journal of Computational and Applied Mathematics》1997,83(1):71-85
In this paper, we propose a new interative and parallel solver, based on domain decomposition, for the h-p version of the finite element method in two dimensions. It improves our previous work in two aspects: (1) A subdomain may contain several super-elements of the coarse mesh, thus can be of arbitrary shape and size. This makes the solver more efficient and more flexible in computational practice. (2) The p-version components (i.e., the high order side and internal modes) in every element are treated separately, which results in better parallelism. 相似文献
2.
Approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces , are given. The results are applied to estimate the rate of convergence when the p-version finite element method is used to approximate the -Laplacian. It is shown that the rate of convergence of the p-version is always at least that of the h-version (measured in terms of number of degrees of freedom used). If the solution is very smooth then the p-version attains an exponential rate of convergence. If the solution has certain types of singularity, the rate of convergence
of the p-version is twice that of the h-version. The analysis generalises the work of Babuska and others to the case . In addition, the approximation theoretic results find immediate application for some types of spectral and spectral element
methods.
Received August 2, 1995 / Revised version received January 26, 1998 相似文献
3.
Luca F. Pavarino 《Numerische Mathematik》1993,66(1):493-515
Summary In this paper, we study some additive Schwarz methods (ASM) for thep-version finite element method. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. We prove a constant bound independent of the degreep and the number of subdomainsN, for the condition number of the ASM iteration operator. This optimal result is obtained first in dimension two. It is then generalized to dimensionn and to a variant of the method on the interface. Numerical experiments confirming these results are reported. As is the case for other additive Schwarz methods, our algorithms are highly parallel and scalable.This work was supported in part by the Applied Math. Sci. Program of the U.S. Department of Energy under contract DE-FG02-88ER25053 and, in part, by the National Science Foundation under Grant NSF-CCR-9204255 相似文献
4.
Summary. In the framework of adaptive methods, bases of hierarchical type are used in the -version of the finite element method. We study the matrices corresponding to the commonly used basis, introduced by Babuška
and Szabo, in the case of -dimensional rectangular elements for 2 order elliptic boundary value problems. For the internal nodes, we show that the condition number is equivalent to and to for the stiffness and mass matrix, respectively. Moreover, we show that the usual diagonal preconditioning divides in the
previous orders the exponents of by two. Finally, we compare these results with those obtained for spectral elements (nodal basis).
Received November 24, 1994 / Revised version received March 20, 1995 相似文献
5.
Barbara I. Wohlmuth 《Numerische Mathematik》1999,84(1):143-171
Summary. A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite
element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves
to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual
based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained
in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably
reduced if the special structure of the Lagrange multiplier is taken into account.
Received July 18, 1997 / Revised version received July 27, 1998 / Published online September 7, 1999 相似文献
6.
Norbert Heuer 《Numerische Mathematik》1998,79(3):371-396
Summary. We study preconditioners for the -version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are
based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis
functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete
natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner.
In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the
mesh size. For all these methods we prove that the resulting condition numbers are bounded by . Here, is the polynomial degree of the ansatz functions and is a constant which is independent of and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported.
Received July 8, 1996 / Revised version received January 8, 1997 相似文献
7.
Domain decomposition iterative procedures for solving scalar waves in the frequency domain 总被引:1,自引:0,他引:1
Seongjai Kim 《Numerische Mathematik》1998,79(2):231-259
The propagation of dispersive waves can be modeled relevantly in the frequency domain. A wave problem in the frequency domain
is difficult to solve numerically. In addition to having a complex–valued solution, the problem is neither Hermitian symmetric
nor coercive in a wide range of applications in Geophysics or Quantum–Mechanics. In this paper, we consider a parallel domain
decomposition iterative procedure for solving the problem by finite differences or conforming finite element methods. The
analysis includes the decomposition of the domain into either the individual elements or larger subdomains ( of finite elements). To accelerate the speed of convergence, we introduce relaxation parameters on the subdomain interfaces
and an artificial damping iteration. The convergence rate of the resulting algorithm turns out to be independent on the mesh
size and the wave number. Numerical results carried out on an nCUBE2 parallel computer are presented to show the effectiveness
of the method.
Received October 30, 1995 / Revised version received January 10, 1997 相似文献
8.
Mark Ainsworth 《Numerische Mathematik》1996,73(4):399-418
Summary.
The element residual method for a posteriori error estimation is analyzed
for degree finite element approximation on quadrilateral elements.
The influence of the choice of subspace used to solve the element residual
problem is studied. It is shown that the resulting estimators will be
consistent (or asymptotically exact) for all
if and only if
the mesh is parallel. Moreover, even if the mesh consists of rectangles, then
the estimators can be inconsistent when .
The results provide concrete guidelines for the selection of a posteriori
error estimators and establish the limits of their performance. In particular,
the use of the element residual method for high orders of approximation
(such as those arising in the - version finite element method) is
vindicated.
The mechanism behind the rather poor performance of the estimators is traced
back to the basic formulation of the residual problem. The investigations
reveal a deficiency in the formulation, leading, as it does, to spurious
modes in the true solution of the residual problem. The recommended choice
of subspaces may be viewed as being sufficient to guarantee that the spurious
modes are filtered out from the approximate solution while at the same time
retaining a sufficient degree of approximation to represent the true modes.
Received February 27, 1995 / Revised version
received June 7, 1995 相似文献
9.
Summary.
We analyze the convergence of a substructuring iterative method
with Lagrange multipliers, proposed recently by Farhat and Roux.
The method decomposes finite element
discretization of an elliptic boundary value problem into
Neumann problems on the subdomains plus a coarse problem for the
subdomain nullspace components. For linear conforming elements and
preconditioning by the Dirichlet problems on the subdomains,
we prove the asymptotic bound on the condition number
,
or ,where
is the characteristic element size and
subdomain size.
Received January 3, 1995 相似文献
10.
We study approximation errors for the h-version of Nédélec edge elements on anisotropically refined meshes in polyhedra. Both tetrahedral and hexahedral elements are considered, and the emphasis is on obtaining optimal convergence rates in the H(curl) norm for higher order elements. Two types of estimates are presented: First, interpolation error estimates for functions in anisotropic weighted Sobolev spaces. Here we consider not only the H(curl)-conforming Nédélec elements, but also the H(div)-conforming Raviart-Thomas elements which appear naturally in the discrete version of the de Rham complex. Our technique is to transport error estimates from the reference element to the physical element via highly anisotropic coordinate transformations. Second, Galerkin error estimates for the standard H(curl) approximation of time harmonic Maxwell equations. Here we use the anisotropic weighted Sobolev regularity of the solution on domains with three-dimensional edges and corners. We also prove the discrete compactness property needed for the convergence of the Maxwell eigenvalue problem. Our results generalize those of [40] to the case of polyhedral corners and higher order elements. 相似文献
11.
A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems 总被引:2,自引:0,他引:2
Summary. We present a Lagrange multiplier based two-level domain decomposition method for solving iteratively large-scale systems
of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed method
is essentially an extension of the regularized FETI (Finite Element Tearing and Interconnecting) method to indefinite problems.
Its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix, and the
preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality
of the residual to a set of carefully chosen planar waves. We show numerically that the proposed method is scalable with respect
to the mesh size, the subdomain size, and the wavenumber. We report performance results for a submarine application that highlight
the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite
elements.
Received March 17, 1998 / Revised version received June 7, 1999 / Published online January 27, 2000 相似文献
12.
Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented. 相似文献
13.
Summary. Additive Schwarz preconditioners are developed for the p-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The principal
preconditioner consists of decomposing the subspace into local spaces associated with the element interiors supplemented with
a wirebasket space associated with the the element interfaces. The wirebasket correction involves inverting a diagonal matrix. If exact
solvers are used on the element interiors then theoretical analysis shows that growth of the condition number of the preconditioned
system is bounded by for an open surface and for a closed surface. A modified form of the preconditioner only requires the inversion of a diagonal matrix but results
in a further degradation of the condition number by a factor .
Received December 15, 1998 / Revised version received March 26, 1999 / Published online March 16, 2000 相似文献
14.
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
15.
Summary.
The interpolation theorem for convex quadrilateral
isoparametric finite elements is proved in the case when the condition
is not satisfied, where is the
diameter of the element and
is the radius of an
inscribed circle in .
The interpolation error is
in the -norm and
in the
-norm provided
that the interpolated function belongs to
. In the case when
the long sides of the quadrilateral
are parallel the constants
appearing in the estimates are evaluated.
Received
September 1993 / Revised version received March 6, 1995 相似文献
16.
Summary.
We estimate condition numbers of -version matrices
for tensor
product elements with two choices of reference element degrees of
freedom. In
one case (Lagrange elements) the condition numbers grow
exponentially in ,
whereas in the other (hierarchical basis functions based on
Tchebycheff
polynomials) the condition numbers grow rapidly but only
algebraically in .
We conjecture that regardless of the choice of basis the
condition numbers
grow like or faster, where is the dimension
of the spatial domain.
Received
August 8, 1992 / Revised version received March 25, 1994 相似文献
17.
Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. In this Note we construct the coarse grid space using the low frequency modes of the subdomain DtN (Dirichlet–Neumann) maps, and apply the obtained two-level preconditioner to the linear system arising from an overlapping domain decomposition. Our method is suitable for the parallel implementation and its efficiency is demonstrated by numerical examples on problems with high heterogeneities. 相似文献
18.
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
19.
Summary This paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting The main emphasis is placed on the analysis when the (exact) solution has singularity of x-type. The first part analyzes thep-version, the second theh-version and generalh-p version and the final third part addresses the problems of the adaptiveh-p version.Supported by the NSF Grant DMS-8315216Partially supported by ONR Contract N00014-85-K-0169 相似文献
20.
We study two-level additive Schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting
from C0 interior penalty methods for fourth order elliptic boundary value problems. We show that the condition number of the preconditioned
system is bounded by C(1+(H3/δ3)), where H is the typical diameter of a subdomain, δ measures the overlap among the subdomains and the positive constant C is independent of the mesh sizes and the number of subdomains.
This work was supported in part by the National Science Foundation under Grant No. DMS-03-11790. 相似文献