共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
Summary. Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems. Received August 9, 1999 / Revised version received May 19, 2000 / Published online March 20, 2001 相似文献
4.
Summary. A semidiscrete mixed finite element approximation to parabolic initial-boundary value problems is introduced and analyzed.
Superconvergence estimates for both pressure and velocity are obtained. The estimates for the errors in pressure and velocity
depend on the smoothness of the initial data including the limiting cases of data in and data in , for sufficiently large. Because of the smoothing properties of the parabolic operator, these estimates for large time levels
essentially coincide with the estimates obtained earlier for smooth solutions. However, for small time intervals we obtain
the correct convergence orders for nonsmooth data.
Received July 30, 1995 / Revised version received October 14, 1996 相似文献
5.
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
6.
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
7.
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
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.
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
10.
Summary. Two new augmented Lagrangian formulations for mixed finite element schemes are presented. The methods lead, in some cases,
to an improvement in the order of the approximation. An error analysis is provided, together with some interesting examples
of applications.
Received July 27, 1994 / Revised version received November 17, 1995 相似文献
11.
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
12.
Nonlinear Galerkin methods and mixed finite elements:
two-grid algorithms for the Navier-Stokes equations 总被引:14,自引:0,他引:14
Summary.
A nonlinear Galerkin method using mixed finite
elements is presented for the two-dimensional
incompressible Navier-Stokes equations. The
scheme is based on two finite element spaces
and for the approximation of the velocity,
defined respectively on one coarse grid with grid
size and one fine grid with grid size and
one finite element space for the approximation
of the pressure. Nonlinearity and time
dependence are both treated on the coarse space.
We prove that the difference between the new
nonlinear Galerkin method and the standard
Galerkin solution is of the order of $H^2$, both in
velocity ( and pressure norm).
We also discuss a penalized version of our algorithm
which enjoys similar properties.
Received October 5, 1993 / Revised version received November
29, 1993 相似文献
13.
Summary. Both for the - and -norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on
the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct
scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation.
Received April 19, 1999 / Published online April 20, 2000 相似文献
14.
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
15.
Summary. Computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model Poisson-problem
in two and three space dimensions. Besides a refined residual-based a posteriori error estimate, an averaging estimator is
established and an -estimate is included. The a posteriori error estimates are reliable and efficient; the proof of reliability relies on a Helmholtz
decomposition.
Received March 4, 1997 / Revised version received September 4, 2001 / Published online December 18, 2001 相似文献
16.
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
17.
Summary. In this paper, we develop and analyze a new finite element method called the sparse finite element method for second order
elliptic problems. This method involves much fewer degrees of freedom than the standard finite element method. We show nevertheless
that such a sparse finite element method still possesses the superconvergence and other high accuracy properties same as those
of the standard finite element method. The main technique in our analysis is the use of some integral identities.
Received October 1, 1995 / Revised version received August 23, 1999 / Published online February 5, 2001 相似文献
18.
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
19.
Summary. Interpolation error estimates for a modified 8-node serendipity finite element are derived in both regular and degenerate
cases, the latter of which includes the case when the element is of triangular shape. For defined over a quadrilateral K, the error for the interpolant is estimated as
, where in the regular case and in the degenerate case, respectively. Thus, the obtained error estimate in the degenerate case is of the same quality as
in the regular case at least for . Results for some related elements are also given.
Received June 2, 1997 / Published online March 16, 2000 相似文献
20.
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