Approximation of Abstract Differential Equations

Approximation of Abstract Differential Equations
D. Guidetti, B. Karasozen and S. Piskarev

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. 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 (see my CV), which contains about 2500 references. Unfortunately, no books or reviews on 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.