共查询到20条相似文献,搜索用时 31 毫秒
1.
We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H
1-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation
operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular
meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in
H
1. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation
which cover both the two- and three-dimensional case in a unified fashion.
相似文献
2.
《国际计算机数学杂志》2012,89(16):2224-2239
In this paper, we investigate the L ∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems. 相似文献
3.
We study two-level additive Schwarz preconditioners for the h-p version of the Galerkin boundary element method when used to solve hypersingular integral equations of the first kind, which
arise from the Neumann problems for the Laplacian in two dimensions. Overlapping and non-overlapping methods are considered.
We prove that the non-overlapping preconditioner yields a system of equations having a condition number bounded by where H
i
is the length of the i-th subdomain, h
i
is the maximum length of the elements in this subdomain, and p is the maximum polynomial degree used. For the overlapping method, we prove that the condition number is bounded by where δ is the size of the overlap and H=max
i
H
i
. We also discuss the use of the non-overlapping method when the mesh is geometrically graded. The condition number in that
case is bounded by clog2
M, where M is the degrees of freedom.
Received October 27, 2000, revised March 26, 2001 相似文献
4.
C. Wieners 《Computing》1997,59(1):29-41
We describe a method for the calculation of theN lowest eigenvalues of fourth-order problems inH
0
2
(Ω). In order to obtain small error bounds, we compute the defects inH
−2(Ω) and, to obtain a bound for the rest of the spectrum, we use a boundary homotopy method. As an example, we compute strict
error bounds (using interval arithmetic to control rounding errors) for the 100 lowest eigenvalues of the clamped plate problem
in the unit square. Applying symmetry properties, we prove the existence of double eigenvalues. 相似文献
5.
One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier–Stokes problem is
the Q
k
−P
k−1
disc
element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version
of the P
k−1
disc
space consisting of piecewise polynomial functions of degree at most k−1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference
cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary
meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that
the latter approach satisfies the inf-sup condition as well for k≥2 in any space dimension.
Received January 31, 2001; revised May 2, 2002 Published online: July 26, 2002 相似文献
6.
This paper is devoted to the a priori error analysis of the hp-version of a streamline-diffusion finite element method for partial differential equations with nonnegative characteristic
form. This class of equations includes second-order elliptic and parabolic problems, first-order hyperbolic problems and second-order
problems of mixed elliptic-parabolic-hyperbolic type. We derive error bounds which are simultaneously optimal in both the
mesh size h and the spectral order p. Numerical examples are presented to confirm the theoretical results.
Received October 28, 1999; revised May 26, 2000 相似文献
7.
Various finite volume element schemes for parabolic integro-differential equations in 1-D are derived and studied. These
types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general
framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowest-order
(linear and L-splines) finite volume elements, although higher-order volume elements can be considered as well under this framework. It
is proved that finite volume element approximations are convergent with optimal order in H
1-norms, suboptimal order in the L
2-norm and super-convergent order in a discrete H
1-norm.
Received August 3, 1998; revised October 11, 1999 相似文献
8.
S. A. Sauter 《Computing》2006,78(2):101-115
It is well known that standard h-version finite element discretisations using lowest order elements for Helmholtz' equation suffer from the following stability
condition: ``The mesh width h of the finite element mesh has to satisfy k
2
h≲1', where k denotes the wave number. This condition rules out the reliable numerical solution of Helmholtz equation in three dimensions
for large wave numbers k≳50. In our paper, we will present a refined finite element theory for highly indefinite Helmholtz problems where the stability
of the discretisation can be checked through an ``almost invariance' condition. As an application, we will consider a one-dimensional
finite element space for the Helmholtz equation and apply our theory to prove stability under the weakened condition hk≲1 and optimal convergence estimates.
Dedicated to Prof. Dr. Ivo Babuška on the occasion of his 80th birthday. 相似文献
9.
This paper investigates a multigrid method for the solution of the saddle point formulation of the discrete Stokes equation
obtained with inf–sup stable nonconforming finite elements of lowest order. A smoother proposed by Braess and Sarazin (1997)
is used and L
2-projection as well as simple averaging are considered as prolongation. The W-cycle convergence in the L
2-norm of the velocity with a rate independently of the level and linearly decreasing with increasing number of smoothing steps
is proven. Numerical tests confirm the theoretically predicted results.
Received January 19, 1999; revised September 13, 1999 相似文献
10.
We consider the streamline-diffusion finite element method with finite elements of lowest order for solving convection-diffusion
problems. Our investigations cover both conforming and nonconforming finite element approximations on triangular and quadrilateral
meshes. Although the considered finite elements are of the same interpolation order their stability and approximation properties
are quite different. We give a detailed overview on the stability and the convergence properties in the L
2- and in the streamline–diffusion norm. Numerical experiments show that often the theoretical predictions on the convergence
properties are sharp.
Received December 7, 1999; revised October 5, 2000 相似文献
11.
Variable Order Panel Clustering 总被引:3,自引:0,他引:3
Stefan Sauter 《Computing》2000,64(3):223-261
We present a new version of the panel clustering method for a sparse representation of boundary integral equations. Instead
of applying the algorithm separately for each matrix row (as in the classical version of the algorithm) we employ more general
block partitionings. Furthermore, a variable order of approximation is used depending on the size of blocks.
We apply this algorithm to a second kind Fredholm integral equation and show that the complexity of the method only depends
linearly on the number, say n, of unknowns. The complexity of the classical matrix oriented approach is O(n
2) while, for the classical panel clustering algorithm, it is O(nlog7
n).
Received July 28, 1999; revised September 21, 1999 相似文献
12.
A Domain Decomposition Preconditioner for p-FEM Discretizations of Two-dimensional Elliptic Problems
S. Beuchler 《Computing》2005,74(4):299-317
In this paper, a uniformly elliptic second order boundary value problem in 2-D discretized by the p-version of the finite element method is considered. An inexact Dirichlet-Dirichlet domain decomposition pre-conditioner for the system of linear algebraic equations is investigated. Two solvers for the problem in the sub-domains, a pre-conditioner for the Schur-complement and an extension operator operating from the edges of the elements into the interior are proposed as ingredients for the inexact DD-pre-conditioner. In the main part of the paper, several numerical experiments on a parallel computer are given. 相似文献
13.
P. Knobloch 《Computing》2006,76(1-2):41-54
We consider a recently introduced triangular nonconforming finite element of third-order accuracy in the energy norm called
Pmod3 element. We show that this finite element is appropriate for approximating the velocity in incompressible flow problems since
it satisfies an inf-sup condition for discontinuous piecewise quadratic pressures. 相似文献
14.
J.-P. Croisille 《Computing》2006,78(4):329-353
We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme
combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil,
which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive
unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L
2 norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on
pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from
1 to 2.5 in the discrete H
1 norm.
This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work. 相似文献
15.
As a first step to developing mathematical support for finite element approximation to the large eddies in fluid motion we
consider herein the Stokes problem. We show that the local average of the usual approximate flow field u
h
over radius δ provides a very accurate approximation to the flow structures of O(δ) or greater. The extra accuracy appears for quadratic or higher velocity elements and degrades to the usual finite element
accuracy as the averaging radius δ→h (the local meshwidth). We give both a priori and a posteriori error estimates incorporating this effect.
Received December 3, 1999; revised October 16, 2000 相似文献
16.
We consider weakly singular integral equations of the first kind on open surface pieces Γ in ℝ3. To obtain approximate solutions we use theh-version Galerkin boundary element method. Furthermore we introduce two-level additive Schwarz operators for non-overlapping
domain decompositions of Γ and we estimate the conditions numbers of these operators with respect to the mesh size. Based
on these operators we derive an a posteriori error estimate for the difference between the exact solution and the Galerkin
solution. The estimate also involves the error which comes from an approximate solution of the Galerkin equations. For uniform
meshes and under the assumption of a saturation condition we show reliability and efficiency of our estimate. Based on this
estimate we introduce an adaptive multilevel algorithm with easily computable local error indicators which allows direction
control of the local refinements. The theoretical results are illustrated by numerical examples for plane and curved surfaces.
Supported by the German Research Foundation (DFG) under grant Ste 238/25-9. 相似文献
17.
S. K. Tomar 《Computing》2006,78(2):117-143
We propose a new h-p spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on
non-smooth domains. The method is shown to be exponentially accurate and asymptotically faster than the standard h-p finite element method. The spectral element functions are fully non-conforming for pure Dirichlet problems and conforming
only at the vertices of the elements for mixed problems, and hence, the dimension of the resulting Schur complement matrix is
quite small. The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method with an almost optimal preconditioner. The algorithm is suitable for a distributed memory parallel computer. The numerical
results of a number of model problems are presented, which confirm the theoretical estimates. 相似文献
18.
In this paper, an H1-Galerkin mixed finite element method is proposed for the 1-D regularized long wave (RLW) equation ut+ux+uux−δuxxt=0. The existence of unique solutions of the semi-discrete and fully discrete H1-Galerkin mixed finite element methods is proved, and optimal error estimates are established. Our method can simultaneously
approximate the scalar unknown and the vector flux effectively, without requiring the LBB consistency condition. Finally,
some numerical results are provided to illustrate the efficacy of our method. 相似文献
19.
J.-P. Croisille 《Computing》2002,68(1):37-63
u ,∇u)=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u). In this paper, we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion
equation cu
x
−ɛu
xx
=f. In the first one, introduced by B. Courbet in [9,10], the numerical average of the diffusive flux is upwinded along the
sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate. It is based onto the
Euler-MacLaurin quadrature formula for the average of the diffusive flux. We emphasize in each case the link with the SUPG
finite element method.
Received June 7, 2001; revised October 2, 2001 相似文献
20.
Gisbert Stoyan 《Computing》2001,67(1):13-33
We explore the prospects of utilizing the decomposition of the function space (H
1
0)
n
(where n=2,3) into three orthogonal subspaces (as introduced by Velte) for the iterative solution of the Stokes problem. It is shown
that Uzawa and Arrow-Hurwitz iterations – after at most two initial steps – can proceed fully in the third, smallest subspace.
For both methods, we also compute optimal iteration parameters. Here, for two-dimensional problems, the lower estimate of
the inf-sup constant by Horgan and Payne proves useful and provides an inclusion of the spectrum of the Schur complement operator
of the Stokes problem.
We further consider the conjugate gradient method in the third Velte subspace and derive a corresponding convergence estimate.
Computational results show the effectiveness of this approach for discretizations which admit a discrete Velte decomposition.
Received June 11, 1999; revised October 27, 2000 相似文献