A posteriori estimates for partial differential equations. (English) Zbl 1162.65001

Repin, Sergey

Radon Series on Computational and Applied Mathematics 4. Berlin: de Gruyter (ISBN 978-3-11-019153-0/hbk). xi, 316 p. (2008).

This is an excellent textbook for a priori estimation for various mathematical problems described by partial differential equations. It can cover a number of numerical methods including the finite element method, although (or rather since) the employed methodology is in principle independent of specific approximation methods.
It begins with the introductory chapter for some mathematical backgrounds in functional-analytical expression. Then it is followed by a concise overview of various a posteriori error estimation methods developed in the last century. The subsequent main chapters of the book are devoted to a new functional approach to a posteriori error control developed in the last decade. The main idea of this approach is as follows: an integral identity that defines a generalized solution is a source of guaranteed and computable error bounds between this solution and any function in the associated energy space. Those bounds can be derived by purely functional-analytic methods without relying on specific features of approximations or numerical methods.
The basic ideas of this approach are presented by means of a simple elliptic problem in Chapter 3, and the following chapters are devoted to various important classes of problems, e.g., diffusion, linear elasticity, incompressible viscous fluids, nonlinear problems described with variational inequalities, evolutional problems.
The text requires a moderate background in functional analysis and the theory of partial differential equations. It will be particularly useful for experts in computational mathematics, and also for students specializing in applied mathematics.

Reviewer: Fumio Kikuchi (Tokyo)

MSC:

65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
65N15	Error bounds for boundary value problems involving PDEs
74S05	Finite element methods applied to problems in solid mechanics
76M10	Finite element methods applied to problems in fluid mechanics
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74B05	Classical linear elasticity
35J25	Boundary value problems for second-order elliptic equations
76D05	Navier-Stokes equations for incompressible viscous fluids
35Q30	Navier-Stokes equations
35J85	Unilateral problems; variational inequalities (elliptic type) (MSC2000)

Keywords:

a posteriori error estimates; numerical methods; textbook; finite element method; guaranteed and computable error bounds; elliptic problem; diffusion; linear elasticity; incompressible viscous fluids; variational inequalities