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1.
C.V. Pao 《Numerische Mathematik》1998,79(2):261-281
This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under
nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison
theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved
upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic
or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical
engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate
the various rates of convergence of the iterations.
Received November 2, 1995 / Revised version received February 10, 1997 相似文献
2.
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
3.
Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations 总被引:24,自引:0,他引:24
M. Zennaro 《Numerische Mathematik》1997,77(4):549-563
Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems
of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to
dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly
more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property.
Received March 29, 1996 / Revised version received August 12, 1996 相似文献
4.
Stefano Serra 《Numerische Mathematik》1999,81(3):461-495
Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned
conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive
in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and,
in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods
exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations
[36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence.
On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric
problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth.
Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis.
Received December 19, 1995 / Revised version received September 15, 1997 相似文献
5.
Summary. The qualocation methods developed in this paper, with spline trial and test spaces, are suitable for classes of boundary
integral equations with convolutional principal part, on smooth closed curves in the plane. Some of the methods are suitable
for all strongly elliptic equations; that is, for equations in which the even symbol part of the operator dominates. Other
methods are suitable when the odd part dominates.
Received December 27, 1996 / Revised version received April 14, 1997 相似文献
6.
Summary. Here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended
from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential
equations of strongly and of oddly elliptic type. The analysis exploits localization in the form of frozen coefficients, pseudohomogeneous
asymptotic symbol representation of classical pseudodifferential operators, a decisive commutator property of the qualocation
projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical
integration of smooth remainders. Numerical examples show the predicted high orders of convergence.
Received January 29, 1998 / Published online: June 29, 1999 相似文献
7.
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
8.
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
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.
Andrea Toselli 《Numerische Mathematik》2000,86(4):733-752
Summary. A two-level overlapping Schwarz method is considered for a Nédélec finite element approximation of 3D Maxwell's equations.
For a fixed relative overlap, the condition number of the method is bounded, independently of the mesh size of the triangulation
and the number of subregions. Our results are obtained with the assumption that the coarse triangulation is quasi-uniform
and, for the Dirichlet problem, that the domain is convex. Our work generalizes well–known results for conforming finite elements
for second order elliptic scalar equations. Numerical results for one and two-level algorithms are also presented.
Received November 11, 1997 / Revised version received May 26, 1999 / Published online June 21, 2000 相似文献
11.
Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear
systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic
and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned
problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The
structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers
increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties
of iterative solvers.
Received August 5, 2000 / Published online June 20, 2001 相似文献
12.
Christoph Pflaum 《Numerische Mathematik》1998,79(1):141-155
A multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive
sparse grids. The multilevel algorithm consists of several V-cycles in - and -direction. A suitable discretization provide that the discrete equation system can be solved in an efficient way. Numerical
experiments show a convergence rate of order for the multilevel algorithm.
Received April 19, 1996 / Revised version received December 9, 1996 相似文献
13.
For a general class of nonlinear (possibly higher index) differential-algebraic equations we show existence and uniqueness
of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the
basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the
computation of regular solutions are developed. Several numerical examples are included.
Received July 4, 1996 / Revised version received May 12, 1997 相似文献
14.
Summary. Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation
of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard
spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved.
A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established.
Received January 5, 2001 / Revised version received August 6, 2001 / Published online December 18, 2001
Correspondence to: C. Schwab 相似文献
15.
S. Maset 《Numerische Mathematik》2002,90(3):555-562
Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear system of delay
differential equations , where . We prove that no Runge-Kutta method preserves asymptotic stability.
Received January 24, 2000 / Revised version received July 19, 2000 / Published online June 7, 2001 相似文献
16.
Least-squares mixed finite element methods
for non-selfadjoint elliptic problems: I. Error estimates
Summary.
A least-squares mixed finite element
method for general second-order non-selfadjoint
elliptic problems in two- and three-dimensional domains
is formulated and analyzed. The finite element spaces for
the primary solution approximation
and the flux approximation
consist of piecewise polynomials of degree
and respectively.
The method is mildly nonconforming on the boundary.
The cases and
are studied.
It is proved that the method is not subject to the LBB-condition.
Optimal - and
-error estimates are derived for
regular finite element partitions.
Numerical experiments, confirming the theoretical rates of
convergence, are presented.
Received
October 15, 1993 / Revised version received August 2, 1994 相似文献
17.
Summary.
This paper presents general algorithms for the parallel
solution of
finite element problems associated with maximal monotone operators of local
type. The latter concept, which is also introduced here, is well suited to
capture the idea that the given operator is the discretization of a differential
operator that may involve nonlinearities and/or constraints as long as those are
of a local nature. Our algorithms are obtained as a combination of known
algorithms for possibly multi-valued maximal monotone operators with appropriate
decompositions of the domain. This work extends a method due to two of the
authors in the single-valued and linear case.
Received April 25, 1994 相似文献
18.
Block parallel iterative methods for the solution of mildly nonlinear systems of equations of the form are studied. Two-stage methods, where the solution of each block is approximated by an inner iteration, are treated. Both
synchronous and asynchronous versions are analyzed, and both pointwise and blockwise convergence theorems provided. The case
where there are overlapping blocks is also considered. The analysis of the asynchronous method when applied to linear systems
includes cases not treated before in the literature.
Received June 5, 1997 / Revised version received December 29, 1997 相似文献
19.
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
20.
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