共查询到20条相似文献,搜索用时 15 毫秒
1.
Summary.
We discuss the effect of cubature errors
when using the Galerkin method for
approximating the solution of Fredholm integral equations in three
dimensions. The accuracy of the cubature method
has to be chosen such that
the error resulting from this further discretization
does not increase the
asymptotic discretization error. We will show that the
asymptotic accuracy
is not influenced provided that polynomials of a certain degree are
integrated exactly by the cubature method. This is done by applying the
Bramble-Hilbert Lemma to the boundary element method.
Received May 24, 1995 相似文献
2.
Alexander Ženíšek 《Numerische Mathematik》1995,71(3):399-417
Summary.
The finite element method for an elliptic equation with discontinuous
coefficients (obtained for the magnetic potential from Maxwell's
equations) is analyzed in the union of closed domains the boundaries
of which form a system of three circles with the same centre.
As the middle domain is very narrow the triangulations obeying
the maximum angle condition are considered. In the case of piecewise
linear trial functions the maximum rate of
convergence in the norm
of the space is proved
under the following conditions:
1. the exact solution
is piecewise of class ;
2. the family of subtriangulations
of the narrow
subdomain satisfies the maximum angle condition
expressed by relation (38). The paper extends the results of [24].
Received
March 8, 1993 / Revised version received November 28, 1994 相似文献
3.
I. Perugia 《Numerische Mathematik》1999,84(2):305-326
Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and
analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship,
while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic
field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements,
and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches
for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A
finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates.
Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999 相似文献
4.
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
5.
Summary.
We consider the finite element approximation of a
non-Newtonian flow, where the viscosity obeys a general law including
the Carreau or power law. For sufficiently regular solutions we prove
energy type error bounds for the velocity and pressure. These bounds
improve on existing results in the literature. A key step in the
analysis is to prove abstract error bounds initially in a quasi-norm,
which naturally arises in degenerate problems of this type.
Received May 25, 1993 / Revised version received January 11, 1994 相似文献
6.
Lie-heng Wang 《Numerische Mathematik》2002,92(4):771-778
Summary. In this paper, we obtain the error bound for any , for the piecewise quadratic finite element approximation to the obstacle problem, without the hypothesis that the free boundary
has finite length (see [3]).
Received October 31, 2000 / Revised version received July 23, 2001 / Published online October 17, 2001
The project was supported by the National Natural Science Foundation of China 相似文献
7.
Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.The work of this author was partially supported by the National Science Foundation under Grant DMS-84-10324 相似文献
8.
Summary.
We present a mixed finite element approximation
of an elliptic problem with degenerate coefficients, arising in the
study of the electromagnetic field in a resonant structure with
cylindrical symmetry. Optimal error bounds are derived.
Received
May 4, 1994 / Revised version received September 27, 1994 相似文献
9.
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
10.
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
11.
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
12.
The Mortar finite element method with Lagrange multipliers 总被引:19,自引:0,他引:19
Faker Ben Belgacem 《Numerische Mathematik》1999,84(2):173-197
Summary. The present paper deals with a variant of a non conforming domain decomposition technique: the mortar finite element method.
In the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices (and the edges in 3D) of subdomains. It
is shown that, written under primal hybrid formulation, the approximation problem, issued from a discretization of a second
order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates
with respect to natural norms. Finally the parallelization advantages consequence of this variant are also addressed.
Received December 1, 1996 / Revised version received November 23, 1998 / Published online September 24, 1999 相似文献
13.
Summary.
The aim of this work is to study a decoupled algorithm of
a fixed point for solving a
finite element (FE) problem for the approximation of viscoelastic
fluid flow obeying an Oldroyd B differential model. The interest for
this algorithm lies in its applications to numerical simulation and
in the cost of computing. Furthermore it is easy to bring this
algorithm into play.
The unknowns
are
the viscoelastic part of the extra stress tensor,
the velocity and
the pressure.
We suppose that the solution
is sufficiently
smooth and small. The approximation
of stress, velocity and pressure are resp.
discontinuous,
continuous,
continuous FE. Upwinding needed for convection of
, is made
by discontinuous FE. The method consists to
solve alternatively a transport equation for the stress,
and a Stokes like problem for velocity and pressure. Previously,
results of existence of the solution for the approximate problem and
error bounds have been obtained using fixed point
techniques with coupled algorithm.
In this paper we show that the mapping of the decoupled
fixed point algorithm is locally (in a neighbourhood of
)
contracting and we obtain existence, unicity (locally) of the solution
of the approximate problem and error bounds.
Received
July 29, 1994 / Revised version received March 13, 1995 相似文献
14.
15.
Summary.
Convergence for the spatial discretization by linear finite
elements of the non-parametric mean curvature flow is proved under natural
regularity assumptions on the continuous solution. Asymptotic convergence is
also obtained for the time derivative which is proportional to mean curvature.
An existence result for the continuous problem in adequate spaces is
included.
Received September 30, 1993 相似文献
16.
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
17.
Summary.
A coupled semilinear elliptic problem modelling an
irreversible, isothermal chemical reaction is introduced, and
discretised using the usual piecewise linear Galerkin finite element
approximation. An interesting feature of the problem is that a reaction order of
less than one gives rise to a "dead core" region. Initially,
one
reactant is assumed to be acting as a catalyst and is kept constant. It
is shown that error bounds previously obtained for a scheme involving
numerical integration can be improved upon by considering a quadratic regularisation
of the nonlinear term.
This technique is then applied to the full coupled problem, and optimal
and error bounds
are proved in the absence of
quadrature. For a scheme involving numerical integration,
bounds similar to those
obtained for the catalyst problem are shown to hold.
Received May 25, 1993 / Revised version received July 5, 1994 相似文献
18.
Stability and analyticity estimates in maximum-norm are shown for spatially discrete finite element approximations based
on simplicial Lagrange elements for the model heat equation with Dirichlet boundary conditions. The bounds are logarithm free
and valid in arbitrary dimension and for arbitrary polynomial degree. The work continues an earlier study by Schatz et al.
[5] in which Neumann boundary conditions were considered.
Received November 1998 / Revised version received August 11, 1999 / Published online July 12, 2000 相似文献
19.
Tomás Chacón Rebollo 《Numerische Mathematik》1998,79(2):283-319
This paper introduces a stabilization technique for Finite Element numerical solution of 2D and 3D incompressible flow problems.
It may be applied to stabilize the discretization of the pressure gradient, and also of any individual operator term such
as the convection, curl or divergence operators, with specific levels of numerical diffusion for each one of them. Its computational
complexity is reduced with respect to usual (residual-based) stabilization techniques. We consider piecewise affine Finite
Elements, for which we obtain optimal error bounds for steady Navier-Stokes and also for generalized Stokes equations (including
convection). We include some numerical experiment in well known 2D test cases, that show its good performances.
Received March 15, 1996 / Revised version received January 17, 1997 相似文献
20.
Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz
problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that
they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related
to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation
of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context,
we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants
have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra
of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given
as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence.
Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000 相似文献