New difference schemes for partial differential equations. (English) Zbl 1060.65055

Operator Theory: Advances and Applications 148. Basel: Birkhäuser (ISBN 3-7643-7054-8/hbk). ix, 443 p. (2004).
The book is devoted to rather hard mathematical problems in the theory of the abstract Cauchy problems in Hilbert and Banach spaces and their difference analogous on uniform grids (a special attention is paid to the case when approximations are considered only with respect to time and only with respect to single-step and two-step difference schemes).
There are two reasons for complications. One can be illustrated on the example of the abstract Cauchy problem (in a Banach space \(E\)) for the differential equations \(v'(t)+A(t)v(t)=f(t)\) (on \([0,T]\)) where \(A(t)\) is a closed linear positive operator with domain \(D\), independent of \(t\) and dence in \(E\). (The chapter 3) The authors study the well posedness (correctness) of the problem on the base of the coercive inequality \[ \|v'\|_{C(E)}+ \|Av\|_{C(E)} \leq M[\|f\|_{C(E)}+\|v_0\|_{D} ] \] and its generalizations (they include the cases of several families of Banach spaces with Holder and \(L_p\) norms). In the case of the heat equation, these inequalities mean that the authors estimate \(v'\) and the derivatives of the second order with respect to the space variables. Of course, such an approach has its known pluses and minuses.
The second reason for complications is connected with the fact that difference approximations with high order of accuracy are investigated. The construction of the schemes is based either on an exact difference scheme (a consequence of the equation) or Taylor’s expansions on two points (these topics are discussed in the details in the first three chapters). The fundamental role of Padé’s rational functions for \(e^{-z}\) should be mentioned.
In Chapters 5 and 6 the similar results are obtained for the elliptic equation \(-v''(t)+A(t)v(t)=f(t)\) (in \(E\)) and for the hyperbolic equation \(v''(t)+Av(t)=f(t)\) (in a Hilbert space \(H\)). Special examples are given for the standard initial-boundary value problems in the case of a bounded domain \(\Omega\) with the smooth boundary. The case of a cube and the usual difference schemes (with respect to space and time) is of special interest.
In Chapter 7 difference schemes for perturbed problems like \(\varepsilon v'(t)+Av(t)=f(t)\) are considered. Uniform a priori estimates are of special value. The final Chapter 8 is really a short Appendix and is devoted to delay parabolic equations.
The monograph will be of interest mainly to specialists and advanced students in several topics of functional analysis, differential and difference equations.


65J10 Numerical solutions to equations with linear operators
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
35K15 Initial value problems for second-order parabolic equations
35L15 Initial value problems for second-order hyperbolic equations
35J25 Boundary value problems for second-order elliptic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs