共查询到20条相似文献,搜索用时 15 毫秒
1.
Susanne C. Brenner. 《Mathematics of Computation》1999,68(226):559-583
We consider the Poisson equation with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When , the rate of convergence to the singular solution in the energy norm is shown to be , and the rate of convergence to the stress intensity factors is shown to be , where is the largest re-entrant angle of the domain and can be arbitrarily small. The cost of the algorithm is . When , the algorithm can be modified so that the convergence rate to the stress intensity factors is . In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be .
2.
Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of
one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped
Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel
versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended
to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are
not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the
design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid,
indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations.
Supported by Communications and Information Technology Ontario (CITO), Canada.
Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational
and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. 相似文献
3.
Tarek P. Mathew 《Numerische Mathematik》1993,65(1):469-492
Summary In this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric, indefinite linear systems arising from mixed finite element discretizations of elliptic problems. The algorithms include Schwarz methods and iterative refinement methods on locally refined grids. The implementation of Schwarz and iterative refinement algorithms have been discussed in part I. A discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived.Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036. 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 NSF Grant ASC 9003002, while the author was a Visiting, Assistant Researcher at UCLA. 相似文献
4.
Luca F. Pavarino 《Numerische Mathematik》1993,66(1):493-515
Summary In this paper, we study some additive Schwarz methods (ASM) for thep-version finite element method. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. We prove a constant bound independent of the degreep and the number of subdomainsN, for the condition number of the ASM iteration operator. This optimal result is obtained first in dimension two. It is then generalized to dimensionn and to a variant of the method on the interface. Numerical experiments confirming these results are reported. As is the case for other additive Schwarz methods, our algorithms are highly parallel and scalable.This work was supported in part by the Applied Math. Sci. Program of the U.S. Department of Energy under contract DE-FG02-88ER25053 and, in part, by the National Science Foundation under Grant NSF-CCR-9204255 相似文献
5.
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
6.
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
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.
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
9.
Ralf Kornhuber 《Numerische Mathematik》1996,72(4):481-499
Summary.
We derive globally convergent multigrid methods
for discrete elliptic
variational inequalities of the second kind
as obtained from
the approximation of related continuous
problems by piecewise linear finite elements.
The coarse grid corrections are computed
from certain obstacle problems.
The actual constraints are fixed by the
preceding nonlinear fine grid smoothing.
This new approach allows the implementation
as a classical V-cycle and preserves
the usual multigrid efficiency.
We give estimates
for the asymptotic convergence rates.
The numerical results indicate a significant improvement
as compared with previous multigrid approaches.
Received
March 26, 1994 / Revised version received September 22, 1994 相似文献
10.
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
11.
We present a sixth-order explicit compact finite difference scheme to solve the three-dimensional (3D) convection-diffusion equation. We first use a multiscale multigrid method to solve the linear systems arising from a 19-point fourth-order discretization scheme to compute the fourth-order solutions on both a coarse grid and a fine grid. Then an operator-based interpolation scheme combined with an extrapolation technique is used to approximate the sixth-order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid-independent convergence rate for solving convection-diffusion equations with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth-order compact (SOC) scheme, compared with the previously published fourth-order compact (FOC) scheme. 相似文献
12.
Sze-Ping Wong 《Numerische Mathematik》1992,62(1):413-438
Summary This work deals with theH
1 condition numbers and the distribution of theB
h
singular values of the preconditioned operators {B
h
–1
A
h
}0, whereA
h
andB
h
are finite element discretizations of second order elliptic operators,A andB respectively.B is also assumed to be self-adjoint and positive definite. For conforming finite elements, Parter and Wong have shown that the singular values cluster in a positive finite interval. Goldstein also has derived results on the spectral distribution ofB
h
–1
A
h
using a different approach. As a generalization of the results of Parter and Wong, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, the singular values also cluster in a positive finite interval. In particular, if the leading part ofB is the same as the leading part ofA, then the singular values cluster about the point {1}. Two specific methods are given as applications of this theory. They are the penalty method of Babuka and the method of nearly zero boundary conditions of Nitsche. Finally, it will be shown that the same results can be proven by an approach generalized from the work of Goldstein.This research was supported by the National Science Foundation under grant number DMS-8913091. 相似文献
13.
In this paper we introduce a variant of the three-field formulation where we use only two sets of variables. Considering,
to fix the ideas, the homogeneous Dirichlet problem for in , our variables are i) the approximations of u in each sub-domain (each on its own grid), and ii) an approximation of u on the skeleton (the union of the interfaces of the sub-domains) on an independent grid (that could often be uniform). The novelty is in
the way to derive, from , the values of each trace of on the boundary of each . We do it by solving an auxiliary problem on each that resembles the mortar method but is more flexible. Under suitable assumptions, quasi-optimal error estimates are proved,
uniformly with respect to the number and size of the subdomains. A preliminary version of the method and of its theoretical
analysis has been presented in Bertoluzza et al. (15th international conference on domain decomposition methods, 2002). 相似文献
14.
Changhua Yu 《Journal of Computational and Applied Mathematics》2011,236(6):1055-1068
In this paper, the semi-discrete and full discrete biquadratic finite volume element schemes based on optimal stress points for a class of parabolic problems are presented. Optimal order error estimates in H1 and L2 norms are derived. In addition, the superconvergences of numerical gradients at optimal stress points are also discussed. A numerical experiment confirms some results of theoretical analysis. 相似文献
15.
Francisco Gaspar F.J. Lisbona C. Rodrigo 《Journal of Computational and Applied Mathematics》2010,234(4):1027-1035
This paper deals with a stencil-based implementation of a geometric multigrid method on semi-structured triangular grids (triangulations obtained by regular refinement of an irregular coarse triangulation) for linear finite element methods. An efficient and elegant procedure to construct these stencils using a reference stencil associated to a canonical hexagon is proposed. Local Fourier Analysis (LFA) is applied to obtain asymptotic convergence estimates. Numerical experiments are presented to illustrate the efficiency of this geometric multigrid algorithm, which is based on a three-color smoother. 相似文献
16.
Summary We make a theoretical study of the application of a standard hierarchical basis multigrid iteration to the convection diffusion equation, discretized using an upwind finite element discretizations. We show behavior that in some respects is similar to the symmetric positive definite case, but in other respects is markedly different. In particular, we find the rate of convergence depends significantly on parameters which measure the strength of the upwinding, and the size of the convection term. Numerical calculations illustrating some of these effects are given.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440. 相似文献
17.
Jin-ping ZengHai-xiong Yu 《Journal of Computational and Applied Mathematics》2012,236(7):1993-2004
We are concerned with the semilinear elliptic problems. We first investigate the L2-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory. 相似文献
18.
PreconditioningP
1 nonconforming finite elements: Condition numbers and singular value distributions
Sze-Ping Wong 《Numerische Mathematik》1993,65(1):253-271
Summary This work deals with the condition numbers and the distribution of theB
h
singular values of the preconditioned operators {B
h
–1
A
h
}0, whereA
h
andB
h
are discretizations of second order elliptic operatorsA andB usingP
1 nonconforming finite elements of Crouzeix and Raviart.B is also assumed to be self-adjoint and positive definite. For conforming finite elements, Parter and Wong have shown that the singular values cluster in a positive finite interval. These reults are being extended to the aforementioned nonconforming finite elements. It will be shown that, for quasiuniform grids, theB
h
singular values are bounded above and below by positive constants which are independent of the grid sizeh. Moreover, they also cluster in a smaller but usually estimable interval. Issues of implementation are also discussed.This research was supported by the National Science Foundation under grant number DMS-8913091 相似文献
19.
Huidong Yang Walter Zulehner 《Journal of Computational and Applied Mathematics》2011,235(18):5367-5379
Fluid-structure interaction problems arise in many fields of application such as flows around elastic structures and blood flow in arteries. The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov-Poincaré operators. This interface equation is solved by a Newton iteration, for which directional derivatives involving shape derivatives with respect to the interface perturbation have to be evaluated appropriately. One step of the Newton iteration requires the solution of several decoupled linear sub-problems in the structure and the fluid domains. These sub-problems are spatially discretized by a finite element method on hybrid meshes. For the time discretization, implicit first-order methods are used for both sub-problems. The discretized equations are solved by algebraic multigrid methods. 相似文献
20.
Summary Subspace decompositions of finite element spaces based onL
2-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors have proved the equivalence of the associated discrete norms with theH
1-norm. The present paper gives an elementary, self-contained derivation of this result which is based on the use ofK-functionals known from the theory of interpolation spaces. 相似文献