The present paper is devoted to the numerical analysis of abstract differential equations in Banach spaces. As is well-known, most of the finite-difference, finite-element, and projection methods can be considered from the point of view of general approximation schemes. 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.
Unfortunately, no books or reviews on general approximation theory have appeared for differential equations in abstract spaces during the last 20 years. Any information on the subject can be found in the original papers only. The paper under review is a first step toward describing a complete picture of discretization methods for abstract differential equations in Banach spaces. It consist of six sections.
In Section 2, the authors 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., 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 the 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 Section 4, which is based on the theory of appproximation of local C-semigroups. Since the backward Cauchy problem is very important in applications and admits stochastic noise, the authors also consider approximation using a stochastic regularization. Such an approach was apparently never before considered in the literature.
In Section 5, discrete coercive inequalities are presented for abstract parabolic equations in various spaces. The last section, Section 6, deals with semilinear problems. The authors consider approximations of Cauchy problems and also problems with periodic solutions. The approach described here is based on the theory of rotation of vector fields and the principle of compact approximation of operators.