Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients

This paper is concerned with obtaining upper and lower bounds for the eigenvalues of Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of composite material mechanics.The problem is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Numerical examples illustrate the accuracy of the results.

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Résumé Dans cet article les bornes supérieures et inférieures sont détermineés pour les valeurs caractéristiques des problèmes de Sturm-Liouville avec des coefficients discontinus. De tels problèmes se trouvent naturellement dans la mécanique des materiaux composites.Après avoir transformé ce problème en utilisant un analogue de la transformation classique de Liouville, les bornes supérieures sont obtenues par l'application d'une technique de Rayleigh-Ritz au problème transformé. Les bornes inférieurs sont determinées en fonction des coefficients sous une forme explicite. Quelques exemples numériques montrent l'exactitude des résultats.

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This work was supported by the U.S. Army Research Office under Grants DAH C04-75-G-0059, DAAG 29-76-G-0063 and DAAG 29-77-G-0034.     

具有不连续系数的奇异摄动拟线性边值问题

研究一类具有不连续系数的奇异摄动二阶拟线性边值问题,其解因一阶导数的不连续性而出现内部层.用合成展开法和上下解定理得到所提问题内部层解的存在性和渐近估计.所得结果应用到由Farrell等（Farrell P A,O＇Riordan E,Shishkin G.A class of singularly perturbed quasilineax differential equations with interiors layers.Mathematics of Computation,2009,78：103-127）所提出的一个特殊拟线性问题.     

A mortar element method for elliptic problems with discontinuous coefficients

This paper proposes a mortar finite element method for solvingthe two-dimensional second-order elliptic problem with jumpsin coefficients across the interface between two subregions.Non-matching finite element grids are allowed on the interface,so independent triangulations can be used in different subregions.Explicitly realizable mortar conditions are introduced to couplethe individual discretizations. The same optimal L²-norm andenergy-norm error estimates as for regular problems are achievedwhen the interface is of arbitrary shape but smooth, thoughthe regularity of the true solution is low in the whole physicaldomain.     

Comparison of two cell-centered multigrid schemes for problems with discontinuous coefficients

Two different schemes for constructing coarse-grid operators are implemented in a linear multigrid code. In the first scheme, the construction of the coarse-grid operators is done using a variational approach. Certain conservation properties of the fine-grid matrices are shown to be preserved on the coarser grids by the variational construction. In the second scheme, the diffusion coefficients for the coarse grids are calculated by a simple restriction of the coefficient from the fine grid, using a flux conservation principle. The multigrid codes are then applied to solve the linear equations from an IMPES formulation of a two-phase porous-media flow model. A standard elliptic model problem with jump discontinuous coefficients is also solved using the two multigrid schemes. In simple cases of particular elliptic equations these two schemes are identical. However, in more general cases, such as in reservoir problems, these schemes differ. It is shown that multigrid efficiency typical of the constant coefficient cases is obtained for these problems involving discontinuous coefficients. © 1993 John Wiley & Sons, Inc.     

The Dirichlet problem for non‐divergence parabolic equations with discontinuous in time coefficients

We consider the Dirichlet problem for non‐divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi‐linear parabolic equations in a bounded domain. In particular, if the boundary is of class C^1,δ , δ ∈ [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dirichlet series with periodic coefficients

In this paper we shall unify the results obtained so far in various scattered literature, for Dirichlet characters and the associated Dirichlet L-functions, under the paradigm of periodic arithmetic functions and the associated Dirichlet series. Notably we shall determine the Laurent coefficients of the series in question to cover Funakura’s result and proceed on to prove the Ayoub-Berndt-Carlitz-Chowla-Müller-Redmond theorem.     

Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients

Summary I give a sufficient condition in order that a Dirichlet problem is solvable in H ² (Ω) for a class of linear second order elliptic partial differential equations. Such a class includes some particular cases for which the result is known.

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Sunto Si prova una condizione sufficiente affinchè un problema di Dirichlet sia risolubile in H ² (Ω) per una classe di equazioni differenziali alle derivate parziali lineari ellittiche del secondo ordine. Tale classe comprende alcuni casi particolari per i quali il risultato è noto.

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The present work was written while the author was a member of the ? Centro di Matematica e Fisica Teorica del C.N.R. ? at the University of Genova, directed by professorJ. Cecconi.

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Entrata in Redazione il 25 febbraio 1971.     

Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions

Summary. Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods. Received April 6, 1994 / Revised version received December 7, 1994     

Some remarks on bounds to eigenvalues of Sturm-Liouville problems with discontinuous coefficients

Upper and lower bounds on the eigenvalues of Sturm-Liouville problems with discontinuous coefficients are discussed. Rayleigh-Ritz approximations based both on Rayleigh's quotient and the dual Rayleigh quotient are used for obtaining upper bounds for the eigenvalues. Though previous studies have indicated that such approximations yield poor results when large discontinuities in the coefficients occur, it is shown in this paper by means of numerical examples that thesame rate of convergence can be achieved as for systems with continuous coefficients, provided the trial functions are allowed to have arbitrary jump discontinuities in their derivatives across the points where the coefficients suffer discontinuities. New explicit lower bounds in terms of the coefficients are also established. The accuracy of the new estimates is illustrated by numerical examples.

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Résumé On discute les bornes supérieures et inférieures des valeurs caractéristiques des problèmes de Sturm-Liouville avec des coefficients discontinus. Les approximations de Rayleigh-Ritz, basées sur le quotient de Rayleigh et le quotient jumelé de Rayleigh, sont utilisées pour obtenir les bornes supérieures des valeurs caractéristiques. Bien que les études antérieures aient indiqué que ces approximations donnent des résultats médiocres quand les coefficients ont de grandes discontinuités, on démontre dans cet article par des exemples numériques qu'on peut réaliser le même degré de convergence que pour les systèmes á coefficients continuous, pourvu que les fonctions d'essai admises aient des sauts arbitraires dans leurs dérivées á travers les points où les coefficients subissent des discontinuités. De nouvelles bornes inférieures sont déterminées sous une forme explicite en fonction des coefficients. On montre l'exactitude des nouveaux résultats par des exemples numériques.

     

Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients

An asymptotic solution of a singularly perturbed linear-quadratic optimal control problem with discontinuous coefficients is constructed by directly substituting an boundary-layer asymptotic expansion of the solution into the condition of the problem and considering a series of problems for finding the asymptotic terms. The error in the approximate solution is estimated. It is shown that the values of the minimized functional do not increase when the next approximations of the optimal control are used.     

Forward-backward SDEs with discontinuous coefficients

In this paper, we are interested in the well-posedness of a class of fully coupled forward-backward SDE (FBSDE) in which the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. Such an FBSDE is motivated by a practical issue in regime-switching term structure interest rate models, and the discontinuity makes it beyond any existing framework of FBSDEs. In a Markovian setting with non-degenerate forward diffusion, we show that a decoupling function can still be constructed and that it is a Sobolev solution to the corresponding quasilinear PDE. As a consequence we can then argue that the FBSDE admits a weak solution in the sense of [1 Antonelli, F., Ma, J. (2003). Weak solutions of forward-backward SDE’s. Stochastic Analysis and Applications 21(3):493–514.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Ma, J., Zhang, J., Zheng, Z. (2008). Weak solutions for backward stochastic differential equations, A martingale approach. The Annals of Probability 36(6):2092–2125.[Crossref], [Web of Science ®] , [Google Scholar]]. In the one-dimensional case, we further prove that the weak solution of the FBSDE is actually strong, and it is pathwisely unique. Our approach does not use the well-known Yamada–Watanabe Theorem, but instead follows the idea of Krylov for SDEs with measurable coefficients.     

Dirichlet and neumann problems for a fractional ordinary differential equation with constant coefficients

For an ordinary differential equation with constant coefficients, we find a general representation of solutions; by using it, we solve the Dirichlet and Neumann problems, obtain conditions for the unique solvability, and construct closed-form representations of solutions.     

Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces

Numerical Algorithms - This paper is concerned with using discontinuous Galerkin isogeometric analysis (dG-IGA) as a numerical treatment of diffusion problems on orientable surfaces ${Omega }...     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

A Neumann-Neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients

Summary. Neumann-Neumann algorithm have been well developed for standard finite element discretization of elliptic problems with discontinuous coefficients. In this paper, an algorithm of this kind is designed and analyzed for a mortar finite element discretization of problems in three dimensions. It is established that its rate of convergence is independent of the discretization parameters and jumps of coefficients between subregions. The algorithm is well suited for parallel computations.Mathematics Subject Classification (1991): 65N55, 65N10, 65N30, 65N22.The work was supported in part by the U.S. Department of Energy under contract DE-FG02-92ER25127 and in part by Polish Science Foundation under grant 2P03A00524.AcknowledgmentThe author would like to thank Olof Widlund for many fruitful discussions and valuable remarks and suggestions on how to improve the presentation of our results.