共查询到20条相似文献,搜索用时 10 毫秒
1.
Sorin Micu 《Numerische Mathematik》2002,91(4):723-768
Summary. A numerical scheme for the controlled semi-discrete 1-D wave equation is considered. We analyze the convergence of the boundary
controls of the semi-discrete equations to a control of the continuous wave equation when the mesh size tends to zero. We
prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded
controls and any weak limit of this sequence is a control for the continuous problem. The number of the eliminated frequencies
depends on the mesh size and the regularity of the continuous initial data. The case of the HUM controls is also discussed.
Received March 3, 2001 / Published online October 17, 2001 相似文献
2.
Keith E. Muller 《Numerische Mathematik》2001,90(1):179-196
Summary. The confluent hypergeometric function, M(a,b,x), arises naturally in both statistics and physics. Although analytically well-behaved, extreme but practically useful combinations
of parameters create extreme computational difficulties. A brief review of known analytic and computational results highlights
some difficult regions, including , with x much larger than b. Existing power series and integral representations may fail to converge numerically, while asymptotic series representations
may diverge before achieving the accuracy desired. Continued fraction representations help somewhat. Variable precision can
circumvent the problem, but with reductions in speed and convenience. In some cases, known analytic properties allow transforming
a difficult computation into an easier one. The combination of existing computational forms and transformations still leaves
gaps. For , two new power series, in terms of Gamma and Beta cumulative distribution functions respectively, help in some cases. Numerical
evaluations highlight the abilities and limitations of existing and new methods. Overall, a rational approximation due to
Luke and the new Gamma-based series provide the best performance.
Received August 16, 1999 / Revised version received September 15, 2000 / Published online May 4, 2001 相似文献
3.
Yunkang Li 《Numerische Mathematik》1995,70(4):473-485
Summary.
This paper deals with the subject of
numerical stability for the neutral functional-differential equation
It is proved that numerical solutions generated by
-methods
are convergent if . However, our numerical experiment suggests
that they are divergent when is large. In
order to obtain convergent numerical solutions when
,
we use
-methods to obtain approximants
to some high order derivative
of the exact solution, then we use the Taylor expansion with integral
remainder to obtain approximants to the exact solution. Since
the equation under consideration has unbounded time lags, it
is in general difficult to investigate numerically the long time
dynamical behaviour of the exact solution due to limited computer
(random access) memory. To avoid this problem we
transform the equation under consideration into a neutral
equation with constant time lags. Using the
later equation as a test model, we prove that the linear
-method
is -stable, i.e., the numerical
solution tends to zero for
any constant stepsize as long as
and , if and only
if , and that the
one-leg -method is -stable if
. We also
find out that inappropriate stepsize causes spurious solution in the
marginal case where and .
Received May 6, 1994 相似文献
4.
On the asymptotic stability properties of Runge-Kutta methods for delay differential equations 总被引:5,自引:0,他引:5
Nicola Guglielmi 《Numerische Mathematik》1997,77(4):467-485
Summary. In this paper asymptotic stability properties of Runge-Kutta (R-K) methods for delay differential equations (DDEs) are considered
with respect to the following test equation: where and is a continuous real-valued function. In the last few years, stability properties of R-K methods applied to DDEs have been
studied by numerous authors who have considered regions of asymptotic stability for “any positive delay” (and thus independent
of the specific value of ).
In this work we direct attention at the dependence of stability regions on a fixed delay . In particular, natural Runge-Kutta methods for DDEs are extensively examined.
Received April 15, 1996 / Revised version received August 8, 1996 相似文献
5.
Arian Novruzi 《Numerische Mathematik》2001,88(1):185-201
Summary. In this paper we establish a error estimation on the boundary for the solution of an exterior Neumann problem in . To solve this problem we consider an integral representation which depends from the solution of a boundary integral equation.
We use a full piecewise linear discretisation which on one hand leads to a simple numerical algorithm but on the other hand
the error analysis becomes more difficult due to the singularity of the integral kernel. We construct a particular approximation
for the solution of the boundary integral equation, for the solution of the Neumann problem and its gradient on the boundary
and estimate their error.
Received May 11, 1998 / Revised version received July 7, 1999 / Published online August 24, 2000 相似文献
6.
Summary. In this paper, we derive the optimal error bounds for the stabilized MITC3 element [3], the MIN3 type element [7] and the T3BL element [8]. In this way we have
solved the problem proposed recently in [5] in a positive manner. Moreover, we estimate the difference between stabilized
MITC3 and MIN3 and show it is of order uniform in the plate thickness.
Received May 31, 2000 / Revised version received April 2, 2001 / Published online September 19, 2001 相似文献
7.
Summary. We study here in detail the location of the real and complex zeros of the partial sums of and , which extends results of Szeg? (1924) and Kappert (1996).
Received November 9, 2000 / Published online August 17, 2001 相似文献
8.
Daisuke Furihata 《Numerische Mathematik》2001,87(4):675-699
Summary. We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes
a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly
ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The
new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation.
The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by
discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate
for the solution is obtained and the order is . Numerical examples demonstrate the effectiveness of the proposed scheme.
Received July 22, 1997 / Revised version received October 19, 1999 / Published online August 2, 2000 相似文献
9.
In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard
finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the
various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.
Received June 11, 1997 / Revised version received December 15, 1997 相似文献
10.
Shinji Adachi Kazunaga Tanaka 《Calculus of Variations and Partial Differential Equations》2000,11(1):63-95
We consider the existence of positive solutions of the following semilinear elliptic problem in :
where , , , and . Under the conditions:
1° for all ,
2° as ,
3° there exist and such that
4°,
we show that (*) has at least four positive solutions for sufficiently small but .
Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000 相似文献
11.
Summary. We combine a primal mixed finite element approach with a Dirichlet-to-Neumann mapping (arising from the boundary integral
equation method) to study the weak solvability and Galerkin approximations of a class of linear exterior transmission problems
in potential theory. Our results are mainly based on the Babuska-Brezzi theory for variational problems with constraints.
We establish the uniqueness of solution for the continuous and discrete formulations, and show that finite element subspac
es of Lagrange type satisfy the discrete compatibility conditions. In addition, we provide the error analysis, including polygonal
approximations of the domain, and prove strong convergence of the Galerkin solutions. Moreover, under additional regularity
assumptions on the solution of the continuous formulation, we obtain the asymptotic rate of convergence O(h).
Received August 25, 1998 / Revised version received March 8, 2000 / Published online October 16, 2000 相似文献
12.
Summary. We propose and analyze a finite difference scheme for the Kohn Laplacian operator associated with the Heisenberg group, which
is a degenerate elliptic operator of H?rmander type. We give a complete analysis for a periodic problem in a cube. In particular,
we prove a discrete Poincaré-Wiertinger inequality which yields the stability. Numerical tests are presented.
Received June 1, 1999 / Revised version received June 7, 2000 / Published online March 20, 2001 相似文献
13.
K. Segeth 《Numerische Mathematik》1999,83(3):455-475
Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown
for a nonlinear parabolic initial-boundary value problem.
Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999 相似文献
14.
Ch. Lubich 《Numerische Mathematik》1994,67(3):365-389
Summary. Convergence estimates in terms of the data are shown for
multistep methods applied to non-homogeneous linear initial-boundary
value problems. Similar error bounds are derived
for a
new class of time-discrete and
fully discrete approximation
schemes for boundary integral equations of such
problems, e.g., for the single-layer potential
equation of the wave equation. In both cases,
the results are obtained from convergence and
stability estimates for operational quadrature
approximations of convolutions.
These estimates, which are also proved here, depend on bounds of the
Laplace transform of the (distributional)
convolution kernel outside the stability region scaled
by the time stepsize, and on the smoothness of the
data.
Received
January 18, 1993 / Revised version received September 15,
1993 相似文献
15.
Isabelle Gallagher 《Numerische Mathematik》2002,91(2):223-236
Summary. We show the consistency and the convergence of a spectral approximation of the bidimensional vorticity equation, proposed
by V. Zeitlin in[13] and studied numerically by I. Szunyogh, B. Kadar, and D. Dévényi in [12], whose main feature is that
it preserves the Hamiltonian structure of the vorticity equation.
Received February 22, 2000 / Revised version received October 23, 2000 / Published online June 20, 2001 相似文献
16.
Summary. In the study of the choice of the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems, Scherzer,
Engl and Kunisch proposed an a posteriori strategy in 1993. To prove the optimality of the strategy, they imposed many very
restrictive conditions on the problem under consideration. Their results are difficult to apply to concrete problems since
one can not make sure whether their assumptions are valid.
In this paper we give a further study on this strategy, and show that Tikhonov regularization is order optimal for each with the regularization parameter chosen according to this strategy under some simple and easy-checking assumptions. This
paper weakens the conditions needed in the existing results, and provides a theoretical guidance to numerical experiments.
Received August 8, 1997 / Revised version received January 26, 1998 相似文献
17.
Hao Lu 《Numerische Mathematik》1997,75(3):339-356
Summary. Galerkin and weighted Galerkin methods are proposed for the numerical solution of parabolic partial differential equations
where the diffusion coefficient takes different signs. The approach is based on a simultaneous discretization of space and
time variables by using continuous finite element methods. Under some simple assumptions, error estimates and some numerical
results for both Galerkin and weighted Galerkin methods are presented. Comparisons with the previous methods show that new
methods not only can be used to solve a wider class of equations but also require less regularity for the solution and need
fewer computations.
Received March 3, 1995 相似文献
18.
Florian Jarre 《Numerische Mathematik》1994,68(1):81-94
Summary. We define the notion of self-concordance of
order two
for the restriction
of a logarithmic barrier function to a
given
line. Based on this notion we prove an inner approximation of the domain
of , as well as a lower bound of the distance from
a point $t$ to the minimum of .
These results provide the theoretical tools to
develop a simple and efficient search step for
finding the minimum of the barrier function along a given line.
The new bound on the size of the line-search step is better than
the optimal bound known
for the case of a self-concordant function (of order one).
We conclude with some numerical examples that
illustrate the promise of the new line-search step.
Received May 24, 1993 / Revised version received February 1994 相似文献
19.
Summary. For evolution equations with a strongly monotone operator we derive unconditional stability and discretization error estimates valid for all . For the -method, with , we prove an error estimate , if , where is the maximal integration step for an arbitrary choice of sequence of steps and with no assumptions about the existence
of the Jacobian as well as other derivatives of the operator , and an optimal estimate under some additional relation between neighboring steps. The first result is an improvement over the implicit midpoint method
, for which an order reduction to sometimes may occur for infinitely stiff problems. Numerical tests illustrate the results.
Received March 10, 1999 / Revised version received April 3, 2000 / Published online February 5, 2001 相似文献
20.
Summary. We consider the heat equation in a smooth domain of R
with Dirichlet and Neumann boundary conditions. It is solved by using its integral formulation with double-layer potentials,
where the unknown , the jump of the solution through the boundary, belongs to an anisotropic Sobolev space. We approximate by the Galerkin method and use a prewavelet basis on , which characterizes the anisotropic space. The use of prewavelets allows to compress the stiffness matrix from to when N is the size of the matrix, and the condition number of the compressed matrix is uniformly bounded as the initial one in the
prewavelet basis. Finally we show that the compressed scheme converges as fast as the Galerkin one, even for the Dirichlet
problem which does not admit a coercive variational formulation.
Received April 16, 1999 / Published online August 2, 2000 相似文献