Sergei Piskarev

Approximation of Abstract Differential Equations

This review paper is devoted to the numerical analysis of abstract differential equations in Banach spaces. Most of the finite difference, finite element, and projection methods can be considered from the point of view of general approximation schemes (see, e.g., [207,210,211] for such a representation). Results obtained for general approximation schemes make the formulation of concrete numerical methods easier and give an overview of methods which are suitable for different classes of problems. The qualitative theory of differential equations in Banach spaces is presented in many brilliant papers and books. We can refer to the bibliography [218], which contains about 3000 references. Unfortunately, no books or reviews on general approximation theory appear for differential equations in abstract spaces during last 20 years. Any information on the subject can be found in the original papers only. It seems that such a review is the first step towards describing a complete picture of discretization methods for abstract differential equations in Banach spaces. In Sec. 2 we describe the general approximation scheme, different types of convergence of operators, and the relation between the convergence and the approximation of spectra. Also, such a convergence analysis can be used if one considers elliptic problems, i.e., the problems which do not depend on time. Section 3 contains a complete picture of the theory of discretization of semigroups on Banach spaces. It summarizes Trotter–Kato and Lax–Richtmyer theorems from the general and common point of view and related problems. The approximation of ill-posed problems is considered in Sec. 4, which is based on the theory of approximation of local C-semigroups. Since the backward Cauchy problem is very important in applications and admits a stochastic noise, we also consider approximation using a stochastic regularization. Such an approach was never considered in the literature before to the best of our knowledge. In Sec. 5, we present discrete coercive inequalities for abstract parabolic equations in Cτn([0, T ];En), C τn([0, T ];En), L p τn([0, T ];En), and Bτn([0, T ];C (Ωh)) spaces.

Semidiscrete approximations of semilinear periodic problems in Banach spaces

where operator A generates C0-semigroup exp(·A). It is well-known [1] that the C0-semigroup gives the solution of (1) by the formula v(t)= exp(tA)v0 for t≥0. We consider the semidiscrete approximation of the problem (1) in the Banach spaces En: v′ n(t)=Anvn(t); t ∈ [0;∞); vn(0)= v n; with v0 n → v0 and the operators An, which generate C0-semigroups and are consistent with the operator A. We understand consistence in the sense of general approximation scheme. This general approximation scheme can be described in the following way [2]. Let En and E be Banach spaces and {pn} be the system of linear bounded operators pn :E→En with the property: ‖pnx‖En →‖x‖E as n→∞ for any x∈E:

On real interpolation, finite differences, and estimates depending on a parameter for discretizations of elliptic boundary value problems

We give some results concerning the real-interpolation method and finite differences. Next, we apply them to estimate the resolvents of finite-difference discretizations of Dirichlet boundary value problems for elliptic equations in space dimensions one and two in analogs of spaces of continuous and Holder continuous functions. Such results were employed to study finite-difference discretizations of parabolic equations.