共查询到20条相似文献,搜索用时 15 毫秒
1.
C.V. Pao 《Numerische Mathematik》1995,72(2):239-262
Summary.
Two block monotone iterative schemes for a nonlinear
algebraic system, which is a finite difference approximation of a
nonlinear elliptic boundary-value problem, are presented and are
shown to converge monotonically either from above or from below to
a solution of the system. This monotone convergence result yields
a computational algorithm for numerical solutions as well as an
existence-comparison theorem of the system, including a sufficient
condition for the uniqueness of the solution. An advantage of the
block iterative schemes is that the Thomas algorithm can be used to
compute numerical solutions of the sequence of iterations in the
same fashion as for one-dimensional problems. The block iterative
schemes are compared with the point monotone iterative schemes of
Picard, Jacobi and Gauss-Seidel, and various theoretical comparison
results among these monotone iterative schemes are given. These
comparison results demonstrate that the sequence of iterations from
the block iterative schemes converges faster than the corresponding
sequence given by the point iterative schemes. Application of the
iterative schemes is given to a logistic model problem in ecology
and numerical ressults for a test problem with known analytical
solution are given.
Received
August 1, 1993 / Revised version received November 7, 1994 相似文献
2.
Summary.
Hybrid methods for the solution of systems of linear equations
consist of a first phase where some information about the associated
coefficient matrix is acquired, and a second phase in which a
polynomial iteration designed with respect to this information is
used. Most of the hybrid algorithms proposed recently for the
solution of nonsymmetric systems rely on the direct use of
eigenvalue estimates constructed by the Arnoldi process in Phase I.
We will show the limitations of this approach and propose an
alternative, also based on the Arnoldi process, which approximates
the field of values of the coefficient matrix and of its inverse in
the Krylov subspace. We also report on numerical experiments
comparing the resulting new method with other hybrid algorithms.
Received May 27, 1993 / Revised version received
November 14, 1994 相似文献
3.
Convergence of block two-stage iterative methods for symmetric positive definite systems 总被引:2,自引:0,他引:2
Zhi-Hao Cao 《Numerische Mathematik》2001,90(1):47-63
Summary. We study the convergence of two-stage iterative methods for solving symmetric positive definite (spd) systems. The main tool
we used to derive the iterative methods and to analyze their convergence is the diagonally compensated reduction (cf. [1]).
Received December 11, 1997 / Revised version received March 25, 1999 / Published online May 30, 2001 相似文献
4.
Asynchronous two-stage iterative methods 总被引:9,自引:0,他引:9
Summary.
Parallel block two-stage iterative methods
for the solution of linear systems of algebraic equations are studied.
Convergence is shown for monotone matrices and for -matrices.
Two different asynchronous versions of these methods
are considered and their convergence investigated.
Received September 7, 1993 / Revised version received April
21, 1994 相似文献
5.
On the convergence of line iterative methods for cyclically
reduced non-symmetrizable linear systems
Summary. We derive analytic bounds on the convergence factors associated
with block relaxation methods for solving the discrete
two-dimensional convection-diffusion equation. The analysis
applies to the reduced systems derived when one step of block
Gaussian elimination is performed on red-black ordered
two-cyclic discretizations. We consider the case where centered
finite difference discretization is used and one cell Reynolds
number is less than one in absolute value and the other is
greater than one. It is shown that line ordered relaxation
exhibits very fast rates of convergence.
Received March 3, 1992/Revised version received July 2, 1993 相似文献
6.
Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods 总被引:3,自引:0,他引:3
Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results
hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are
examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different
cases where the matrix is monotone, singular, and positive (semi)definite are studied.
Received September 5, 1995 / Revised version received April 3, 1996 相似文献
7.
Summary. We consider a second-order elliptic equation with discontinuous or anisotropic coefficients in a bounded two- or three dimensional domain, and its finite-element discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients. Received February 5, 1999 / Published online March 16, 2000 相似文献
8.
Summary.
A coupled semilinear elliptic problem modelling an
irreversible, isothermal chemical reaction is introduced, and
discretised using the usual piecewise linear Galerkin finite element
approximation. An interesting feature of the problem is that a reaction order of
less than one gives rise to a "dead core" region. Initially,
one
reactant is assumed to be acting as a catalyst and is kept constant. It
is shown that error bounds previously obtained for a scheme involving
numerical integration can be improved upon by considering a quadratic regularisation
of the nonlinear term.
This technique is then applied to the full coupled problem, and optimal
and error bounds
are proved in the absence of
quadrature. For a scheme involving numerical integration,
bounds similar to those
obtained for the catalyst problem are shown to hold.
Received May 25, 1993 / Revised version received July 5, 1994 相似文献
9.
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
10.
In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their
discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete
flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting
the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas
vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence
free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite
element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient
spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof.
Received November 4, 1996 / Revised version received February 2, 1998 相似文献
11.
Nonlinear Galerkin methods and mixed finite elements:
two-grid algorithms for the Navier-Stokes equations 总被引:14,自引:0,他引:14
Summary.
A nonlinear Galerkin method using mixed finite
elements is presented for the two-dimensional
incompressible Navier-Stokes equations. The
scheme is based on two finite element spaces
and for the approximation of the velocity,
defined respectively on one coarse grid with grid
size and one fine grid with grid size and
one finite element space for the approximation
of the pressure. Nonlinearity and time
dependence are both treated on the coarse space.
We prove that the difference between the new
nonlinear Galerkin method and the standard
Galerkin solution is of the order of $H^2$, both in
velocity ( and pressure norm).
We also discuss a penalized version of our algorithm
which enjoys similar properties.
Received October 5, 1993 / Revised version received November
29, 1993 相似文献
12.
Summary.
An adaptive finite element method for the calculation of transonic potential
flows was developed. An error indicator based on first order finite differences
of gradients is introduced as a local error estimator. It measures second order
distributional derivatives. Estimates involving
this error estimator, a residual and the error are given. The error estimator
can be used as a criterion for mesh refinement. We also give some computational
results.
Received September 16, 1993 / Revised version received June
7, 1994 相似文献
13.
Summary. The aim of this paper is to prove some Babuška–Brezzi type conditions which are involved in the mortar spectral element discretization
of the Stokes problem, for several cases of nonconforming domain decompositions.
ID=" <E5>Dedicated to Olof B. Widlund on the occasion of his 60th birthday</E5> 相似文献
14.
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
15.
Summary. We discuss a finite difference preconditioner for the interpolatory cubic spline collocation method for a uniformly elliptic operator defined by in (the unit square) with homogeneous Dirichlet boundary conditions. Using the generalized field of values arguments, we discuss
the eigenvalues of the preconditioned matrix where is the matrix of the collocation discretization operator corresponding to , and is the matrix of the finite difference operator corresponding to the uniformly elliptic operator given by in with homogeneous Dirichlet boundary conditions. Finally we mention a bound of -singular values of for a general elliptic operator in .
Received December 11, 1995 / Revised version received June 20, 1996 相似文献
16.
Summary.
Large, sparse nonsymmetric systems of linear equations with a
matrix whose eigenvalues lie in the right half plane may be solved by an
iterative method based on Chebyshev polynomials for an interval in the
complex plane. Knowledge of the convex hull of the spectrum of the
matrix is required in order to choose parameters upon which the
iteration depends. Adaptive Chebyshev algorithms, in which these
parameters are determined by using eigenvalue estimates computed by the
power method or modifications thereof, have been described by Manteuffel
[18]. This paper presents an adaptive Chebyshev iterative method, in
which eigenvalue estimates are computed from modified moments determined
during the iterations. The computation of eigenvalue estimates from
modified moments requires less computer storage than when eigenvalue
estimates are computed by a power method and yields faster convergence
for many problems.
Received May 13, 1992/Revised version received May 13,
1993 相似文献
17.
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
18.
The topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered
finite volume method. Dirichlet boundary conditions are considered here. A discrete Poincaré inequality is used, and estimates
on the approximate solutions are proven. The convergence of the scheme without any assumption on the regularity of the exact
solution is proven using some compactness results which are shown to hold for the approximate solutions.
Received January 16, 1998 / Revised version received June 19, 1998 相似文献
19.
Monotone iterative methods for finite difference system of reaction-diffusion equations 总被引:6,自引:0,他引:6
C. V. Pao 《Numerische Mathematik》1985,46(4):571-586
Summary This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences. 相似文献
20.
A fully discrete finite element method is used to approximate the electric field equation derived from time-dependent Maxwell's equations in three dimensional polyhedral domains. Optimal energy-norm error estimates are achieved for general Lipschitz polyhedral domains. Optimal -norm error estimates are obtained for convex polyhedral domains. Received February 3, 1997 / Revised version received February 27, 1998 相似文献