On the best approximations and rate of convergence of decompositions in the root vectors of an operator

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator A that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

Solvability for p-Laplacian generalized fractional coupled systems with two-sided memory effects

This paper determines the solvability of multipoint boundary value problems for p-Laplacian generalized fractional differential systems with Riesz–Caputo derivative, which exhibits two-sided nonlocal memory effects. An equivalent integral form for the generalized fractional differential system is deduced by transformation. First, we obtain the existence of solutions on the basis of the upper–lower solutions method, in which an explicit iterative approach for approximating the solution is established. Second, we deal with a special case of our fractional differential system; in order to obtain novel results, an abstract sum-type operator equation A(x,x)+Bx+e=x on ordered Banach space is discussed. Without requiring the existence of upper–lower solutions or compactness conditions, we get several unique results of solutions for this operator equation, which provide new inspiration for the study of boundary value problems. Then, we apply these abstract results to get the uniqueness of solutions for our differential system.     

Approximation methods and alternative problems

Projection methods applied to abstract problems of the form Ax = Nx, where A is a linear operator with a nontrivial null space and N is a nonlinear operator, both on a normed space, are studied. Convergence results are obtained and then are applied to periodic two-point boundary value problems using splines as approximations.     

Existence results for evolutionary inclusions and variational–hemivariational inequalities

We consider an abstract first-order evolutionary inclusion in a reflexive Banach space. The inclusion contains the sum of L-pseudomonotone operator and a maximal monotone operator. We provide an existence theorem which is a generalization of former results known in the literature. Next, we apply our result to the case of nonlinear variational–hemivariational inequalities considered in the setting of an evolution triple of spaces. We specify the multivalued operators in the problem and obtain existence results for several classes of variational–hemivariational inequality problems. Finally, we illustrate our existence result and treat a class of quasilinear parabolic problems under nonmonotone and multivalued flux boundary conditions.     

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.     

Error estimates for successive approximations and spectral properties of linear operators

The aim of this paper is to describe some relations between the convergence speed of successive approximations to solutions of linear operator equations, on the one hand, and various spectral properties of the corresponding operators, on the other. We shall show, in particular, that the estimates for the convergence speed of successive approximations is basically determined by certain properties of the pheripheral spectrum of the operator involved (recall that the peripheral spectrum is that part of the spectrum which lies on the boundary, i.e. consists of numbers with absolute values equal to the spectral radius). Equivalently, the convergence speed is characterized by the growth of the (Fredholm) resolvent when approaching the peripheral spectrum. Interestingly, these properties are essentially different for Volterra and non-Volterra operators, where by Volterra operator we mean, as usual, an operator whose spectrum consists only of zero.     

Approximation of a fractional power of an elliptic operator

Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present article, new integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two‐dimensional boundary value problem for fractional diffusion.     

Asymptotics for 2‐D wave equations with Wentzell boundary conditions in the square

This paper concerns the asymptotics of the linear wave equation with frictional damping only on Wentzell boundary in the square. After reformulating the model into an abstract Cauchy problem, we show that the spectrum for the corresponding operator matrix has no purely imaginary values. Moreover, by analyzing a family of eigenvalues for the operator matrix, we prove that there exists a solution of the system, whose energy decay rate can be arbitrarily slow.     

Transmutation of non-local boundary conditions in ocean acoustics

We employ a transmutation technique to construct a new non-local boundary condition for the paraxial approximation in ocean acoustics. The transmutation operator introduced by De Santo and Polyanskii, when applied to the Helmholtz equation governing the acoustic pressure in the water column, leads to the so-called parabolic equation of Fock and Tappert. This transmutation operator acting on the N-D map at the water–bottom interface yields an intermediate non-local boundary condition for the parabolic equation which eliminates the backscattering terms in the Schwartz kernel of the N-D map. The kernel of the intermediate condition is approximated by a uniform stationary phase formula taking account of the possible coalescence of the brach points of the integrand with the stationary points of the phase, and it leads to a non-local boundary condition of Volterra-type for the parabolic equation. This condition is quite different than similar conditions derived by other approximations, in that the kernel of the Volterra operator is smooth, the smoothing effect coming from the fact that the horizontal range coordinate is scaled with the relative refraction index between the water column and the bottom.     

Stability for Lie–Trotter products for some operator matrix semigroups

Coupled systems of linear differential equations in Banach spaces can be often handled by the theory of C₀-semigroups of operator matrices. We study the stability of Lie–Trotter products of such matrix semigroups, and present three classes of examples (abstract delay equations, abstract inhomogeneous equations, abstract dynamic boundary value problems) and some open problems. This survey is based on the papers [1], [2] and [5], to which we refer the interested reader for more details and extensive bibliographical information. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resolvent Krylov subspace approximation to operator functions

We consider the approximation of operator functions in resolvent Krylov subspaces. Besides many other applications, such approximations are currently of high interest for the approximation of φ-functions that arise in the numerical solution of evolution equations by exponential integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behaviour if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we analyse a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators whose field of values lies in the left half plane. In contrast to standard Krylov methods, the convergence will be independent of the norm of the discretised operator and thus of the spatial discretisation. We will discuss efficient implementations for finite element discretisations and illustrate our analysis with numerical experiments.     

Vers des lois de parois multi-échelle implicites

The purpose of this Note is to present a unifying approach of boundary layer approximations for the Laplace operator in domains with periodic rugous boundaries. We show a negative result for an averaged second-order like wall-law. To circumvent this difficulty, we propose new multi-scale wall-laws that include microscopic oscillations on the fictitious boundary. In a first step they are explicit non-homogeneous Dirichlet conditions, afterwards an implicit multi-scale Saffman–Joseph-like wall-law is derived. We establish theoretical orders of convergence and provide their numerical assessment, as well as a counter-example that demonstrates the impossibility of a real averaged second order wall-law. To cite this article: D. Bresch, V. Milisic, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Abstract green formula and applications to boundary integral equations

The resolution of boundary value problems by integral equations is usually based on isomorphisms between the solution of the boundary value problem and boundary data. Using an abstract Green formula in a Hilbert space framework, we prove these isomorphisms. Many applications are given, like the Dirichlet and Neumann problems for the Laplace operator, as well as the clamped and free plate problems in the plane.     

A twist condition and multiple solutions of unbounded self-adjoint operator equation with symmetries

By using the index theory for unbounded self-adjoint operator equations and the symmetric mountain pass theorem, we investigate the existence of multiple solutions for nonlinear operator equations with twist conditions. We prove an abstract theorem, and give some applications to first order Hamiltonian systems with Sturm–Liouville boundary conditions and delay differential equations.     

Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

The paper defines the Weyl–Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livšic. Examples of boundary value problems for partial differential equations and calculations of their Weyl–Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl–Titchmarsh function for the three dimensional Schr?dinger operator introduced by W.O. Amrein and D.B. Pearson. Relationships to the Schr?dinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted. Received: May 21, 2007., Revised: October 8, 2007., Accepted: October 11, 2007     

Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator

We study an abstract nonlinear evolution equation governed by a time-dependent operator of subdifferential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.     

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

Goursat problem for two-dimensional second-order hyperbolic operator-differential equations with variable domains

We develop a modification of the energy inequality method and use it to prove the well-posedness of the Goursat problem for linear second-order hyperbolic differential equations with operator coefficients whose domains depend on the two-dimensional time. An energy inequality for strong solutions of this Goursat problem is derived with the help of abstract smoothing operators, and we prove that the range of the problem is dense by using properties of a regularizing Cauchy problem whose inverse operator is a family of smoothing operators of a new type. We give an example of a well-posed boundary value problem for a two-dimensional complete second-order hyperbolic partial differential equation with Goursat conditions and with a boundary condition depending on the two-dimensional time.     

Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.