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1.
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
2.
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
3.
Summary.
We consider two level overlapping Schwarz domain decomposition methods
for solving the finite element problems that arise from
discretizations of elliptic problems on general unstructured meshes
in two and three dimensions. Standard finite element interpolation
from
the coarse to the fine grid may be used. Our theory requires no
assumption on the substructures
that constitute the whole domain, so the
substructures can be of arbitrary shape and of different
size. The global coarse mesh is allowed to be non-nested
to the fine grid on which the discrete problem is to be solved, and
neither
the coarse mesh nor the fine mesh need be quasi-uniform.
In addition, the domains defined by the fine and coarse grid need
not be identical. The one important constraint is that the closure
of the coarse grid must cover any portion of the fine grid boundary
for which Neumann boundary conditions are given.
In this general setting, our algorithms have the same optimal
convergence rate as the usual two level overlapping domain decomposition
methods on structured meshes.
The condition number of the preconditioned system depends only on the
(possibly small)
overlap of the
substructures and the size of the coarse grid, but is independent of
the sizes of the subdomains.
Received
March 23, 1994 / Revised version received June 2, 1995 相似文献
4.
Tarek P. Mathew 《Numerische Mathematik》1993,65(1):445-468
Summary We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by the Army Research Office under Grant DAAL 03-91-G-0150, while the author was a Visiting Assistant Researcher at UCLA 相似文献
5.
We consider a scalar advection-diffusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear system, we propose and study an additive, two-level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions.
6.
We consider a stationary Stokes problem with a piecewise constant viscosity coefficient. For the variational formulation of
this problem we prove a well-posedness result in which the constants are uniform with respect to the jump in the viscosity
coefficient. We apply a standard discretization with a pair of LBB stable finite element spaces. The main result of the paper
is an infsup result for the discrete problem that is uniform with respect to the jump in the viscosity coefficient. From this
we derive a robust estimate for the discretization error. We prove that the mass matrix with respect to some suitable scalar
product yields a robust preconditioner for the Schur complement. Results of numerical experiments are presented that illustrate
this robustness property.
This author was supported by the German Research Foundation through the guest program of SFB 540 相似文献
7.
Summary. We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational
problems posed in the Hilbert spaces and in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz
smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results
are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products.
Received June 12, 1998 / Revised version received March 12, 1999 / Published online January 27, 2000 相似文献
8.
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
9.
We analyze two‐level overlapping Schwarz domain decomposition methods for vector‐valued piecewise linear finite element discretizations of the PDE system of linear elasticity. The focus of our study lies in the application to compressible, particle‐reinforced composites in 3D with large jumps in their material coefficients. We present coefficient‐explicit bounds for the condition number of the two‐level additive Schwarz preconditioned linear system. Thereby, we do not require that the coefficients are resolved by the coarse mesh. The bounds show a dependence of the condition number on the energy of the coarse basis functions, the coarse mesh, and the overlap parameters, as well as the coefficient variation. Similar estimates have been developed for scalar elliptic PDEs by Graham et al. 1 The coarse spaces to which they apply here are assumed to contain the rigid body modes and can be considered as generalizations of the space of piecewise linear vector‐valued functions on a coarse triangulation. The developed estimates provide a concept for the construction of coarse spaces, which can lead to preconditioners that are robust with respect to high contrasts in Young's modulus and the Poisson ratio of the underlying composite. To confirm the sharpness of the theoretical findings, we present numerical results in 3D using vector‐valued linear, multiscale finite element and energy‐minimizing coarse spaces. The theory is not restricted to the isotropic (Lamé) case, extends to the full‐tensor case, and allows applications to multiphase materials with anisotropic constituents in two and three spatial dimensions. However, the bounds will depend on the ratio of largest to smallest eigenvalue of the elasticity tensor. 相似文献
10.
Summary.
In this paper we introduce a class of robust multilevel
interface solvers for two-dimensional
finite element discrete elliptic problems with highly
varying coefficients corresponding to geometric decompositions by a
tensor product of strongly non-uniform meshes.
The global iterations convergence rate is shown to be of
the order
with respect to the number of degrees
of freedom on the single subdomain boundaries, uniformly upon the
coarse and fine mesh sizes, jumps in the coefficients
and aspect ratios of substructures.
As the first approach, we adapt the frequency filtering techniques
[28] to construct robust smoothers
on the highly non-uniform coarse grid. As an alternative, a multilevel
averaging procedure for successive coarse grid correction is
proposed and analyzed.
The resultant multilevel coarse grid
preconditioner is shown to have (in a two level case) the condition
number independent
of the coarse mesh grading and
jumps in the coefficients related to the coarsest refinement level.
The proposed technique exhibited high serial and parallel
performance in the skin diffusion processes modelling [20]
where the high dimensional coarse mesh problem inherits a strong geometrical
and coefficients anisotropy.
The approach may be also applied to magnetostatics problems
as well as in some composite materials simulation.
Received December 27, 1994 相似文献
11.
We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type. 相似文献
12.
L. Ghezzi 《Journal of Computational and Applied Mathematics》2010,234(5):1492-1504
Overlapping Schwarz preconditioners are constructed and numerically studied for Gauss-Lobatto-Legendre (GLL) spectral element discretizations of heterogeneous elliptic problems on nonstandard domains defined by Gordon-Hall transfinite mappings. The results of several test problems in the plane show that the proposed preconditioners retain the good convergence properties of overlapping Schwarz preconditioners for standard affine GLL spectral elements, i.e. their convergence rate is independent of the number of subdomains, of the spectral degree in the case of generous overlap and of the discontinuity jumps in the coefficients of the elliptic operator, while in the case of small overlap, the convergence rate depends on the inverse of the overlap size. 相似文献
13.
Arnold Reusken 《Numerische Mathematik》1995,71(3):365-397
Summary.
We consider a two-grid method for solving 2D convection-diffusion
problems. The coarse grid correction is based on approximation of
the Schur complement. As a preconditioner of the Schur complement we use the
exact Schur complement of modified fine grid equations. We assume constant
coefficients and periodic boundary conditions and apply Fourier analysis. We
prove an upper bound for the spectral radius of the two-grid iteration
matrix that is smaller than one and independent of the mesh size, the
convection/diffusion ratio and the flow direction; i.e. we have a (strong)
robustness result. Numerical results illustrating the robustness of the
corresponding multigrid -cycle are given.
Received October 14, 1994 相似文献
14.
Susanne C. Brenner 《Numerische Mathematik》1996,72(4):419-447
Summary.
A two-level additive Schwarz preconditioner is
developed for the
systems resulting from the discretizations of
the plate bending problem by the Morley finite element, the
Fraeijs de Veubeke finite element, the Zienkiewicz finite
element and the Adini
finite element. The condition numbers of the preconditioned
systems are shown
to be bounded independent of mesh sizes and the number of
subdomains in the
case of a generous overlap.
Received
February 1, 1994 / Revised version received October 24, 1994 相似文献
15.
Summary. Additive Schwarz preconditioners are developed for the p-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The principal
preconditioner consists of decomposing the subspace into local spaces associated with the element interiors supplemented with
a wirebasket space associated with the the element interfaces. The wirebasket correction involves inverting a diagonal matrix. If exact
solvers are used on the element interiors then theoretical analysis shows that growth of the condition number of the preconditioned
system is bounded by for an open surface and for a closed surface. A modified form of the preconditioner only requires the inversion of a diagonal matrix but results
in a further degradation of the condition number by a factor .
Received December 15, 1998 / Revised version received March 26, 1999 / Published online March 16, 2000 相似文献
16.
We study two-level additive Schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting
from C0 interior penalty methods for fourth order elliptic boundary value problems. We show that the condition number of the preconditioned
system is bounded by C(1+(H3/δ3)), where H is the typical diameter of a subdomain, δ measures the overlap among the subdomains and the positive constant C is independent of the mesh sizes and the number of subdomains.
This work was supported in part by the National Science Foundation under Grant No. DMS-03-11790. 相似文献
17.
Mario A. Casarin 《Numerische Mathematik》2001,89(2):307-339
Summary. The - spectral element discretization of the Stokes equation gives rise to an ill-conditioned, indefinite, symmetric linear system
for the velocity and pressure degrees of freedom. We propose a domain decomposition method which involves the solution of
a low-order global, and several local problems, related to the vertices, edges, and interiors of the subdomains. The original
system is reduced to a symmetric equation for the velocity, which can be solved with the conjugate gradient method. We prove
that the condition number of the iteration operator is bounded from above by , where C is a positive constant independent of the degree N and the number of subdomains, and is the inf-sup condition of the pair -. We also consider the stationary Navier-Stokes equations; in each Newton step, a non-symmetric indefinite problem is solved
using a Schwarz preconditioner. By using an especially designed low-order global space, and the solution of local problems
analogous to those decribed above for the Stokes equation, we are able to present a complete theory for the method. We prove
that the number of iterations of the GMRES method, at each Newton step, is bounded from above by . The constant C does not depend on the number of subdomains or N, and it does not deteriorate as the Newton iteration proceeds.
Received March 2, 1998 / Revised version received October 12, 1999 / Published online March 20, 2001 相似文献
18.
Talal Rahman 《Numerical Algorithms》2011,58(2):235-260
We propose a simple and effective hybrid (multiplicative) Schwarz precondtioner for solving systems of algebraic equations
resulting from the mortar finite element discretization of second order elliptic problems on nonmatching meshes. The preconditioner
is embedded in a variant of the classical preconditioned conjugate gradient (PCG) for an effective implementation reducing
the cost of computing the matrix-vector multiplication in each iteration of the PCG. In fact, it serves as a framework for
effective implementation of a class of hybrid Schwarz preconditioners. The preconditioners of this class are based on solving
a sequence of non-overlapping local subproblems exactly, and the coarse problems either exactly or inexactly (approximately).
The classical PCG algorithm is reformulated in order to make reuse of the results of matrix-vector multiplications that are
already available from the preconditioning step resulting in an algorithm which is cost effective. An analysis of the proposed
preconditioner, with numerical results, showing scalability with respect to the number of subdomains, and a convergence which
is independent of the jumps of the coefficients are given. 相似文献
19.
1.引言设0是RZ中的有界多边形区域,其边界为Rfl.考虑下面的重调和Dirichlet问题:(1.1)的变分形式为:求。EHI(fi)使得对?/EL‘(m,问题(1.幻的唯一可解性可由冯(m上的M线性型的强制性和连续性以及La。Mlgram定理得出(of[4]).令人一{丸)是n的一个三角剖分,并且满足最小角条件,其中h是它的网格参数.设Vh为Money元空间[41.问题(1.2)的有限元离散问题为:求。eVh使得当有限元参数人很小时,这个方程组很大,而且矩阵A的条件数变得非常大,直接求解,存贮量及计算量都很大.如果B可逆,则方程组(1.4)等… 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