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61.
This paper presents a set of sufficient conditions for a sequence of semimartingales to converge weakly to a solution of a stochastic differential equation (SDE) with discontinuous drift and diffusion coefficients. This result is closely related to a well-known weak-convergence theorem due to Liptser and Shiryayev (see [27]) which proves the weak convergence to a solution of a SDE with continuous drift and diffusion coefficients in the Skorokhod–Lindvall J
1-topology.The goal of this paper is to obtain a stronger result in order to solve outstanding problems in the area of large-scale queueing networks – in which the weak convergence of normalized queueing length is a solution of a SDE with discontinuous coefficients. To do this we need to make the stronger assumptions: (1) replacing the convergence in probability of the triplets of a sequence of semimartingales in the original Liptser and Shiryayev's theorem by stronger convergence in L
2, (2) assuming the diffusion coefficient is coercive, and (3) assuming the discontinuity sets of the coefficients of the limit diffusion processs are of Lebesgue measure zero. 相似文献
62.
We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.
63.
Locking-free DGFEM for elasticity problems in polygons 总被引:1,自引:0,他引:1
The h-version of the discontinuous Galerkin finite element method(h-DGFEM) for nearly incompressible linear elasticity problemsin polygons is analysed. It is proved that the scheme is robust(locking-free) with respect to volume locking, even in the absenceof H2-regularity of the solution. Furthermore, it is shown thatan appropriate choice of the finite element meshes leads torobust and optimal algebraic convergence rates of the DGFEMeven if the exact solutions do not belong to H2. 相似文献
64.
We consider a discontinuous weight Sturm-Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. We extend and generalize some approaches and results of the classic regular Sturm-Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation in such a way that the problem under consideration can be interpreted as an eigenvalue problem for a suitable selfadjoint operator, construct the Green’s function and resolvent operator, and derive asymptotic formulas for eigenvalues and normalized eigenfunctions.Original Russian Text Copyright © 2005 Mukhtarov O. Sh. and Kadakal M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 860–875, July–August, 2005. 相似文献
65.
We consider a multigrid algorithm (MG) for the cell centered finite difference scheme (CCFD) on general triangular meshes using a new prolongation operator. This prolongation is designed to solve the diffusion equation with strongly discontinuous coefficient as well as with smooth one. We compare our new prolongation with the natural injection and the weighted operator in Kwak, Kwon, and Lee ( 8 ) and the behaviors of these three prolongation are discussed. Numerical experiments show that (i) for smooth problems, the multigrid with our new prolongation is fastest, the next is the weighted prolongation, and the third is the natural injection; and (ii) for nonsmooth problems, our new prolongation is again fastest, the next is the natural injection, and the third is the weighted prolongation. In conclusion, our new prolongation works better than the natural injection and the weighted operator for both smooth and nonsmooth problems. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
66.
研究系数密集间断的边值问题,采用降维方法,使得在数值计算上可实现并行计算,并且收敛性的证明变得很简洁. 相似文献
67.
V. V. Filippov 《Mathematical Notes》1997,61(3):340-345
The verification of an appropriately stated equicontinuity condition for a sequence of solution spaces is one of the two key
points in the theory of the Cauchy problem for equations with singular right-hand sides. We obtain a related sufficient condition.
Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 416–423, March, 1997.
Translated by V. E. Nazaikinskii 相似文献
68.
Kun Li Ting-Zhu Huang Liang Li Stéphane Lanteri 《Numerical Methods for Partial Differential Equations》2023,39(2):932-954
In this artice, we report on a reduced-order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3-D time-domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time-domain (DGTD) solver. With a Galerkin projection and a second order leap-frog (LF2) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD-based ROM and assess its efficiency for 3-D nanophotonic problems. 相似文献
69.
Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients
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Haijun Wu & Weiying Zheng 《数学研究通讯:英文版》2023,39(3):437-475
The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are
considered. Under the conditions that the coefficient is quasi-monotone up to
a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard
multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on
new nodes and their immediate neighbours. The multigrid V-cycle algorithm
uses $\mathcal{O}(N)$ operations per iteration and is optimal. 相似文献
70.
JérÔme Jaffré 《Transport in Porous Media》1995,21(3):195-207
When simulating two-phase flow in porous media, one has to consider the case where there is a discontinuity in the medium. There relative permeabilities and capillary pressure functions may change and we address the problem of calculating the convective part of the numerical flux at the interface between the two rock types. Several solutions are compared. 相似文献