A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

Error analysis for optimal control problem governed by convection diffusion equations: DG method

In the current paper, we study the convergence properties of the DGFE approximation of optimal control problem governed by convection-diffusion equations. We derive a posteriori error estimates and a priori error estimates for both the states, ad-joint and the control variable approximation. For the optimal control problem, these estimates are apparently not available in the literature.     

On the strong stability of differential equations and difference schemes

Paper deals with the construction of a priori estimates for the solution of Cauchy problem for an abstract linear differential equation in Hilbert space under perturbations of the initial condition, right-hand side, and operators of the problem. It is shown that a priori estimates of strong stability can be obtained directly on the basis of various a priori estimates for the solution of the Cauchy problem. The perturbations of the operators of the problem are estimated in the corresponding operator norms. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

STABILIZED NUMERICAL APPROXIMATIONS OF THE BACKWARD PROBLEM OF A PARABOLIC EQUATION

1　IntroductionLetΩ be a bounded domain in Rn and Ω be its boundary.ThenΣ =Ω× ( 0 ,1 ) is abounded domain in Rn+1 .We consider the following backwad problem of a prabolic equa-tion: u t= ni,j=1 xiaij( x) u xj -c( x) u,　　 ( x,t)∈Σ,( 1 )u| Ω× [0 ,1 ] =0 , ( 2 )u| t=1 =g( x) . ( 3 )　　 Where { aij( x) } are smooth functions given onΩ satisfyingaij( x) =aji( x) ,　　 1≤ i,j≤ n, ( 4)α0 ni=1ζ2i ≤ ni,j=1aij( x)ζiζj≤α1 ni=1ζ2i,　　 ζ∈ Rn,x∈Ω. ( 5)　　Where0 <α…     

VARIATIONAL DISCRETIZATION OF A CONTROL-CONSTRAINED PARABOLIC BANG-BANG OPTIMAL CONTROL PROBLEM

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional.For the numerical treatment,we use variational discretization of its Tikhonov regularization:For the state and the adjoint equation,we apply Petrov-Galerkin schemes in time and usual conforming finite elements in space.We prove a-priori estimates for the error between the discretized regularized problem and the limit problem.Since these estimates are not robust if the regularization parameter tends to zero,we establish robust estimates,which--depending on the problem's regularity——enhance the previous ones.In the special case of bang-bang solutions,these estimates are further improved.A numerical example confirms our analytical findings.     

Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem

We consider the higher order Lidstone boundary value problem. New upper and lower estimates for positive solutions of the problem are obtained. A discussion of the sharpness of the estimates is included.     

Meromorphic functions of matrix arguments and applications

Sharp estimates are established for the Euclidean norm of matrix-valued meromorphic functions of a matrix argument. Applications of the obtained estimates to the periodic problem and two-point boundary value problem for vector differential equations are also discussed.     

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.     

Subspace Interpolation with Applications to Elliptic Regularity

In this paper, we prove new embedding results by means of subspace interpolation theory and apply them to establishing regularity estimates for the biharmonic Dirichlet problem and for the Stokes and the Navier–Stokes systems on polygonal domains. The main result of the paper gives a stability estimate for the biharmonic problem at the threshold index of smoothness. The classic regularity estimates for the biharmonic problem are deduced as a simple corollary of the main result. The subspace interpolation tools and techniques presented in this paper can be applied to establishing sharp regularity estimates for other elliptic boundary value problems on polygonal domains.     

Norm estimates for functions of matrices with simple spectrum

Sharp estimates are established for the norm of a matrix-valued function of a matrix having geometrically simple eigenvalues. Applications of the obtained estimates to the periodic problem and two-point boundary value problem for vector differential equations are also discussed.     

A posteriori error estimates for the generalized Stokes problem

The paper concerns a posteriori estimates of functional type for the difference between exact and approximate solutions to a generalized Stokes problem. The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem using the method suggested by the first author. The estimates obtained can be classified into two types. Estimates of the first type are valid only for solenoidal functions, while estimates of the second type are applicable for any functions that belong to the energy space of the respective problem and satisfy the boundary conditions. In the second case, the estimates include an additional penalty term with a multiplier defined by the constant in the Ladyzhenskaya-Babuška-Brezzi condition. It is proved that a posteriori estimates for the velocity field yield computable estimates of the difference between exact and approximate pressure functions in the L₂-norm. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires to solve only finite-dimensional problems. Bibliography: 34 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 89–101.     

Difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients

A nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients is considered. For solving this problem, a priori estimates in the differential and difference forms are obtained. The a priori estimates imply the uniqueness and stability of the solution on a layer with respect to the initial data and the right-hand side and the convergence of the solution of the difference problem to the solution of the differential problem.     

Consistent two-sided estimates for the solutions of quasilinear parabolic equations and their approximations

For a linearized finite-difference scheme approximating the Dirichlet problem for a multidimensional quasilinear parabolic equation with unbounded nonlinearity, we establish pointwise two-sided solution estimates consistent with similar estimates for the differential problem. These estimates are used to prove the convergence of finite-difference schemes in the grid L ₂ norm.     

带有可测系数的二阶非线性抛物型方程组的初—混合边值问题

在多连通区域上研究带有可测系数的二阶非线性抛物型方程组的初－混合边值问题，首先我们将其化为复形式的方程组，并给出在一定条件下的上述初－这值问题解的先验估计，然后利用解的估计和列紧性原理，证明了这种初－边值问题解的存在性。在论证过程中，我们始终用复分析方法讨论文中所提出的问题，没有看到国外有人使用这种方法处理此类问题。     

Integral estimates of lengths of level lines of rational functions and zolotarev’s problem

Integral estimates of lengths of level lines (lemniscates) of rational functions of a complex variable are obtained. These estimates are related to the problem of separation of compact sets by rational functions and to Zolotarev’s problem.     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

A Posteriori Estimates in Local Norms

We derive a posteriori estimates for the difference between exact solutions and approximate solutions to boundary-value problems in terms of local norms. The diffusion problem, linear elasticity and generalizations to other boundary-value elliptic problems are considered. Computable estimates for the deviation from the exact solution are also obtained in terms of linear functionals. Unlike published works of other authors, the construction of such estimates is not connected with any analysis of the adjoint boundary-value problem. On the basis of multiplicative inequalities, local estimates in certain norms subject to the energy norm are derived. Bibliography: 10 titles.     

Upper estimates for the energy of solutions of nonhomogeneous boundary value problems

We establish upper bounds for the energy of critical levels of the functional associated to a perturbed superlinear elliptic boundary value problem. We show that the perturbed problem satisfies the estimates obtained by Bahri and Lions (1988) for the symmetric problem. We use these estimates to prove the existence of nonradial solutions to a radial elliptic boundary value problem. Our results fill a gap in an earlier paper by Aduén and Castro.

     

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.     

混合有限元法的误差分析

关于混合变分问题有限元方法的研究工作,见[1]—[4].其中已得出混合法的最优误差估计.本文讨论抽象混合有限元法的误差并证明一些超收敛估计,然后将其应用到具体问题上,即应用到一个四阶边值问题和一个二阶边值问题.