共查询到20条相似文献,搜索用时 0 毫秒
1.
Joachim Schöberl 《Numerische Mathematik》1999,84(1):97-119
Summary. In this paper we consider multigrid methods for the parameter dependent problem of nearly incompressible materials. We construct
and analyze multilevel-projection algorithms, which can be applied to the mixed as well as to the equivalent, non-conforming
finite element scheme in primal variables. For proper norms, we prove that the smoothing property and the approximation property
hold with constants that are independent of the small parameter. Thus we obtain robust and optimal convergence rates for the
W-cycle and the variable V-cycle multigrid methods. The numerical results pretty well conform the robustness and optimality
of the multigrid methods proposed.
Received June 17, 1998 / Revised version received October 26, 1998 / Published online September 7, 1999 相似文献
2.
Summary. We describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems
with weightedextended b-splines (web-splines). This new technique provides high accuracy with relatively low-dimensional subspaces, does not require
any grid generation, and is ideally suited for hierarchical solution techniques. In particular, we show that the standard
W-cycle yields uniform convergence, i.e., the required number of iterations is bounded independent of the grid width.
Received August 17, 2000 / Published online August 17, 2001 相似文献
3.
Patrick Lacoste 《Numerische Mathematik》2000,84(4):577-609
Summary. This study deals with the mathematical and numerical solution of time-harmonic Maxwell equation in axisymmetric geometry.
Using Fourier decomposition, we define weighted Sobolev spaces of solution and we prove expected regularity results. A practical
contribution of this paper is the construction of a class of finite element conforming with the H (rot) space equipped with the weighted measure rdrdz. It appears as an extension of the well-known cartesian mixed finite element of Raviart-Thomas-Nédélec [11]–[15]. These elements
are built from classical lagrangian and mixed finite element, therefore no special approximations functions are needed. Finally,
following works of Mercier and Raugel [10], we perform an interpolation error estimate for the simplest proposed element.
Received March 15, 1996 / Revised version received November 30, 1998 / Published online December 6, 1999 相似文献
4.
Summary. We analyze V–cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence
properties of the multigrid algorithm of the original problem. As an application, we study the convergence of multigrid algorithms
for a covolume method or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal
domains. As in standard finite element methods, the V–cycle algorithm with one pre-smoothing converges with a rate independent of the number of levels. Various types of smoothers
including point or line Jacobi, and Gauss-Seidel relaxation are considered.
Received August 19, 1999 / Revised version received July 10, 2000 / Published online June 7, 2001 相似文献
5.
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
6.
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
7.
Rob Stevenson 《Numerische Mathematik》1997,78(2):269-303
Summary. In this paper, we introduce a multi-level direct sum space decomposition of general, possibly locally refined linear or multi-linear
finite element spaces. The resulting additive Schwarz preconditioner is optimal for symmetric second order elliptic problems.
Moreover, it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh, perturbations with
positive zeroth order terms, and, after a further decomposition of the spaces, also with respect to anisotropy along the grid
lines. Important for an efficient implementation is that stable bases of the subspaces defining our decomposition, consisting
of functions having small supports can be easily constructed.
Received September 8, 1995 / Revised version received October 31, 1996 相似文献
8.
Summary. In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number
of the preconditioned system is shown to be bounded by , where H measures the diameters of the subdomains, h is the mesh size of the triangulation, and the constant C is independent of H, h and the number of subdomains.
Received August 14, 1997 相似文献
9.
Summary. Three iterative domain decomposition methods are considered: simultaneous updates on all subdomains (Additive Schwarz Method),
flow directed sweeps and double sweeps. By using some techniques of formal language theory we obtain a unique criterion
of convergence for the three methods. The convergence rate is a function of the criterion and depends on the algorithm.
Received October 24, 1994 / Revised version received November 27, 1995 相似文献
10.
This paper deals with a posteriori estimates for the finite element solution of the Stokes problem in stream function and vorticity formulation. For two different
discretizations, we propose error indicators and we prove estimates in order to compare them with the local error. In a second
step, these results are extended to the Navier-Stokes equations.
Received March 25, 1996 / Revised version received April 7, 1997 相似文献
11.
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
12.
Summary. Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in
order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment
of non-homogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the
setting of space refinement (i.e. adaptive) discretizations of second order problems. Our method is based on the use of compatible
multiscale decompositions for both the domain and its boundary, and on the possibility of characterizing various function
spaces from the numerical properties of these decompositions. In particular, this allows the construction of a lifting operator
which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis.
An explicit construction of the wavelet bases and the lifting is proposed on fairly general domains, based on conforming domain decomposition techniques.
Received November 2, 1998 / Published online April 20, 2000 相似文献
13.
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
14.
Summary. A new finite element method for elliptic problems with locally periodic microstructure of length is developed and analyzed. It is shown that the method converges, as , to the solution of the homogenized problem with optimal order in and exponentially in the number of degrees of freedom independent of . The computational work of the method is bounded independently of . Numerical experiments demonstrate the feasibility and confirm the theoretical results. Received September 11, 1998 / Published online April 20, 2000 相似文献
15.
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
16.
Summary. We present a new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry
of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions.
In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the non-density
of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside
the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves
to be numerically efficient.
Received April 27, 2001 / Revised version received September 13, 2001 / Published online March 8, 2002 相似文献
17.
Summary Robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in
the convergence. Non conforming grids make the grid generation much easier and faster since it is then a parallel task. The
goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of Robin interface
conditions with non-matching grids. We consider both a symmetric definite positive operator and the convection-diffusion equation
discretized by finite volume schemes. Numerical results are shown.
Received December 22, 1999 / Revised version received December 21, 2000 / Published online December 18, 2001
Correspondence to: F. Nataf 相似文献
18.
Superconvergence analysis and error expansion for the Wilson nonconforming finite element 总被引:8,自引:0,他引:8
Summary.
In this paper the Wilson nonconforming finite element is considered for
solving a class of two-dimensional second-order elliptic boundary value
problems. Superconvergence estimates and error expansions are obtained
for both uniform and non-uniform rectangular meshes. A new lower bound
of the error shows that the usual error estimates are optimal. Finally
a discussion on the error behaviour in negative norms shows that there
is generally no improvement in the order by going to weaker norms.
Received July 5, 1993 相似文献
19.
20.
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