共查询到20条相似文献,搜索用时 10 毫秒
1.
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
2.
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
3.
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
4.
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
5.
Summary. For the simulation of biomolecular systems in an aqueous solvent a continuum model is often used for the solvent. The accurate
evaluation of the so-called solvation energy coming from the electrostatic interaction between the solute and the surrounding
water molecules is the main issue in this paper. In these simulations, we deal with a potential problem with jumping coefficients
and with a known boundary condition at infinity. One of the advanced ways to solve the problem is to use a multigrid method
on locally refined grids around the solute molecule. In this paper, we focus on the error analysis of the numerical solution
and the numerical solvation energy obtained on the locally refined grids. Based on a rigorous error analysis via a discrete
approximation of the Greens function, we show how to construct the composite grid, to discretize the discontinuity of the
diffusion coefficient and to interpolate the solutions at interfaces between the fine and coarse grids. The error analysis
developed is confirmed by numerical experiments.
Received June 25, 1998 / Revised version received July 14, 1999 / Published online June 8, 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.
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
8.
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
9.
On the use of rational iterations and domain decomposition methods for the Helmholtz problem 总被引:2,自引:0,他引:2
Seongjai Kim 《Numerische Mathematik》1998,79(4):529-552
An iterative algorithm for the numerical solution of the Helmholtz problem is considered. It is difficult to solve the problem
numerically, in particular, when the imaginary part of the wave number is zero or small. We develop a parallel iterative algorithm
based on a rational iteration and a nonoverlapping domain decomposition method for such a non-Hermitian, non-coercive problem.
Algorithm parameters (artificial damping and relaxation) are introduced to accelerate the convergence speed of the iteration.
Convergence analysis and effective strategies for finding efficient algorithm parameters are presented. Numerical results
carried out on an nCUBE2 are given to show the efficiency of the algorithm. To reduce the boundary reflection, we employ a
hybrid absorbing boundary condition (ABC) which combines the first-order ABC and the physical
$Q$
ABC. Computational results comparing the hybrid ABC with non-hybrid ones are presented.
Received May 19, 1994 / Revised version received March 25, 1997 相似文献
10.
Summary. In the Dual-Primal FETI method, introduced by Farhat et al. [5], the domain is decomposed into non-overlapping subdomains,
but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the
remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated.
The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number,
assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded
by for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for
a wide class of finite elements for the Reissner-Mindlin plate model.
Received January 20, 2000 / Revised version received April 25, 2000 / Published online December 19, 2000 相似文献
11.
Silvia Bertoluzza 《Numerische Mathematik》2003,93(4):611-634
Summary. In this paper we prove that, for suitable choices of the bilinear form involved in the stabilization procedure, the stabilized
three fields domain decomposition method proposed in [8] is stable and convergent uniformly in the number of subdomains and with respect to their sizes under quite general assumptions on the decomposition and on the discretization spaces. The same is proven to hold for the
resulting discrete Steklov-Poincaré operator.
Received April 4, 2000 / Revised version received January 9, 2001 / Published online June 17, 2002 相似文献
12.
13.
Norbert Heuer 《Numerische Mathematik》2001,88(3):485-511
Summary. We analyze an additive Schwarz preconditioner for the p-version of the boundary element method for the single layer potential operator on a plane screen in the three-dimensional
Euclidean space. We decompose the ansatz space, which consists of piecewise polynomials of degree p on a mesh of size h, by introducing a coarse mesh of size . After subtraction of the coarse subspace of piecewise constant functions on the coarse mesh this results in local subspaces
of piecewise polynomials living only on elements of size H. This decomposition yields a preconditioner which bounds the spectral condition number of the stiffness matrix by . Numerical results supporting the theory are presented.
Received August 15, 1998 / Revised version received November 11, 1999 / Published online December 19, 2000 相似文献
14.
Summary. Efficiency of high-order essentially non-oscillatory (ENO) approximations of conservation laws can be drastically improved
if ideas of multiresolution analysis are taken into account. These methods of data compression not only reduce the necessary
amount of discrete data but can also serve as tools in detecting local low-dimensional features in the numerical solution.
We describe the mathematical background of the generalized multiresolution analysis as developed by Abgrall and Harten in
[14], [15] and [3]. We were able to ultimately reduce the functional analytic background to matrix-vector operations of linear
algebra. We consider the example of interpolation on the line as well as the important case of multiresolution analysis of
cell average data which is used in finite volume approximations. In contrast to Abgrall and Harten, we develop a robust agglomeration
procedure and recovery algorithms based on least-squeare polynomials. The efficiency of our algorithms is documented by means
of several examples.
Received April 4, 1998 / Revised version August 2, 1999 / Published online June 8, 2000 相似文献
15.
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
16.
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
17.
Summary. In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process
new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing
steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and
positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples.
Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic
problems are presented. They indicate a very robust convergence behavior of the MLILU method.
Received June 13, 1997 / Revised version received March 17, 1998 相似文献
18.
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
19.
Summary. Lower bounds for the condition numbers of the preconditioned systems are obtained for the wire basket preconditioner and
the Neumann-Neumann preconditioner in three dimensions. They show that the known upper bounds are sharp.
Received January 28, 2001 / Revised version received September 3, 2001 / Published online January 30, 2002
This work was supported in part by the National Science Foundation under Grant Nos. DMS-9600133 and DMS-0074246 相似文献
20.
Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first
kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz
operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation
of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators.
Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000 相似文献