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1.
Analysis of FETI methods for multiscale PDEs 总被引:2,自引:0,他引:2
In this paper, we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic
PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain
interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient
function the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h))2 where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if varies only mildly in a layer Ω
i,η
of width η near the boundary of each of the subdomains Ω
i
, then , independent of the variation of α in the remainder Ω
i
\Ω
i,η
of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains.
Our theoretical findings are confirmed in numerical tests.
C. Pechstein was supported by the Austrian Science Funds (FWF) under grant F1306. 相似文献
2.
We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods
in the case that only available data is a noise of y satisfying with a given small noise level . We terminate the iteration by the discrepancy principle in which the stopping index is determined as the first integer such that
with a given number τ > 1. Under certain conditions on {α
k
}, {g
α
} and F, we prove that converges to as and establish various order optimal convergence rate results. It is remarkable that we even can show the order optimality
under merely the Lipschitz condition on the Fréchet derivative F′ of F if is smooth enough. 相似文献
3.
David Walnut 《Journal of Fourier Analysis and Applications》1998,4(6):669-709
Explicit, compactly supported solutions, {vi, ϕ}
i=1
m
, to the deconvolution (or Bezout) equation
are computed where ϕ is a given function in C
c
∞
(Rd), and
, i=1, ..., m for some set of positive numbers {ri}
i=1
m
such that ri/rj is poorly approximated by rationals whenever i ≠ j. The novelty of the solution technique is that it uses new results in
the theory of sampling of bandlimited functions detailed in [13] to provide simple Fourier series representations for the
solutions, {vi, ϕ}
i=1
m
, which can be easily implemented numerically.
Several examples illustrating the use of sampling for solutions to variants of (0.1) are given, as well as some numerical
simulations.
Acknowledgements and Notes. The author gratefully acknowledges the support of the National Science Foundation, DMS-9500909, and Prof. K.J.R. Liu at
the Institute for Systems Research, University of Maryland, College Park, MD, 20742. 相似文献
((0.1)) |
4.
An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson
system in the two- dimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution
of the adaptive mesh from one time step to the next one, based on a rigorous analysis of the local regularity and how it gets
transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step, in the L
∞ metric. The numerical solutions are proved to converge in L
∞ towards the exact ones as ε and Δt tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs
to . The rate of convergence is , which should be compared to the results of Besse who recently established in (SIAM J Numer Anal 42(1):350–382, 2004) similar
rates for a uniform semi-Lagrangian scheme, but requiring that the initial data are in . Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations. 相似文献
5.
Zhaoli Liu Jiabao Su Zhi-Qiang Wang 《Calculus of Variations and Partial Differential Equations》2009,35(4):463-480
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f (x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I
m
is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity
on the asymptotic behaviors of the nonlinearity f and .
Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013)
and the Doctoral Programme Foundation of NEM of China (20070028004). 相似文献
6.
J. Casado-Díaz T. Chacón Rebollo V. Girault M. Gómez Mármol F. Murat 《Numerische Mathematik》2007,105(3):337-374
In this paper we consider, in dimension d≥ 2, the standard finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L
∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an
hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs
to L
1(Ω), we prove that the unique solution of the discrete problem converges in (for every q with ) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon
measure. In the case where the dimension is d = 2 or d = 3 and where the coefficients are smooth, we give an error estimate in when the right-hand side belongs to L
r
(Ω) for some r > 1. 相似文献
7.
Let be such that each is a signed measure on R
d
belonging to the Kato class K
d, 1. A Brownian motion in R
d
with drift is a diffusion process in R
d
whose generator can be informally written as . When each is given by U
i
(x)dx for some function U
i
, a Brownian motion with drift is a diffusion in R
d
with generator . In Kim and Song (Ill J Math 50(3):635–688, 2006), some properties of Brownian motions with measure-value drifts in bounded
smooth domains were discussed. In this paper we prove a scale invariant boundary Harnack principle for the positive harmonic
functions of Brownian motions with measure-value drifts in bounded Lipschitz domains. We also show that the Martin boundary
and the minimal Martin boundary with respect to Brownian motions with measure-valued drifts coincide with the Euclidean boundary
for bounded Lipschitz domains. The results of this paper are also true for diffusions with measure-valued drifts, that is,
when is replaced by a uniformly elliptic divergence form operator with C
1 coefficients or a uniformly elliptic non-divergence form operator with C
1 coefficients.
The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. The research of P. Kim is supported
by Research Settlement Fund for the new faculty of Seoul National University. 相似文献
8.
Let (,G, U) be a continuous representation of a Lie groupG by bounded operatorsg U (g) on the Banach space and let (,
,dU) denote the representation of the Lie algebra
obtained by differentiation. Ifa
1, ...,a
d
is a Lie algebra basis of
,A
i
=dU (a
i
) and
whenever =(i
1, ...,i
k
) we reconsider the operators
相似文献
9.
For Au = f with an elliptic differential operator and stochastic data f, the m-point correlation function of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A
(m) of A. Sparse tensor products of hierarchic FE-spaces in are known to allow for approximations to which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl
Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here
how to achieve the fast computation of sparse approximations of for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method. 相似文献
10.
Pigong Han Zhaoxia Liu 《Calculus of Variations and Partial Differential Equations》2007,30(3):315-352
Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem
11.
Xianling Fan Shao-Gao Deng 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):255-271
We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving
the p(x)-Laplacian of the form
12.
We analyse degenerate, second-order, elliptic operators H in divergence form on L
2(R
n
× R
m
). We assume the coefficients are real symmetric and a
1
H
δ
≥ H ≥ a
2
H
δ
for some a
1, a
2 > 0 where
13.
Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several
conditions on u
0 to prove the existence of a unique strong solution with u(0) = u
0 in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to
prove the following optimal result with the weakest possible initial value condition and the largest possible solution class:
Given u
0, q, s as above and the Stokes operator A
2, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions. 相似文献
14.
It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class
() and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol
of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms
of pseudodifferential operators with -bounded symbols, yielding by an iteration argument the -boundedness of λ(A−λ)−1 in for some . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with -bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential
operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on with operator valued coefficients. 相似文献
15.
Konrad Gröger Lutz Recke 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(3):263-285
This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type
16.
This paper concerns variational inclusions of the form where f is a single locally Lipschitz subanalytic function and F is a set-valued map acting in Banach spaces. We prove the existence and the convergence of a sequence (x
k
) satisfying where lies to which is the Clarke Jacobian of f at the point x
k
.
相似文献
17.
Vladislav Kargin 《Probability Theory and Related Fields》2007,139(3-4):397-413
Let X
i
denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X
i
it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX
i
= 1, then is between and c
2
n for certain constant c
1 and c
2. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X
i
, and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables. 相似文献
18.
Claudie Hassenforder Sabine Mercier 《Annals of the Institute of Statistical Mathematics》2007,59(4):741-755
Let
be a sequence of letters taken in a finite alphabet Θ. Let
be a scoring function and
the corresponding score sequence where X
i
= s(A
i
). The local score is defined as follows:
. We provide the exact distribution of the local score in random sequences in several models. We will first consider a Markov
model on the score sequence
, and then on the letter sequence
. The exact P-value of the local score obtained with both models are compared thanks to several datasets. They are also compared with previous
results using the independent model. 相似文献
19.
20.
Marek Omelka 《Annals of the Institute of Statistical Mathematics》2007,59(2):385-402
The rank statistic
, with R
i
(t) being the rank of
and e
1
, . . . , e
n
being the random sample from a distribution with a cdf F, is considered as a random process with t in the role of parameter. Under some assumptions on c
i
, x
i
and on the underlying distribution, it is proved that the process
converges weakly to the Gaussian process. This generalizes the existing results where the one-dimensional case
was considered. We believe our method of the proof can be easily modified for the signed-rank statistics of Wilcoxon type.
Finally, we use our results to find the second order asymptotic distribution of the R-estimator based on the Wilcoxon scores and also to investigate the length of the confidence interval for a single parameter
β
l
. 相似文献
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