Numerical solution of elliptic and parabolic partial differential equations (Q2904669)

By its deepness, richness and quality of work this is a convincing, great textbook on elliptic (and parabolic) equations with a stress on finite element methods, embracing both the theory (existence, uniqueness, smoothness) of these equations and their (initial) boundary value problems, and the theory of their numerical solution methods. There are even remarks on the implementation of the methods and an enclosed program package.NEWLINENEWLINEThe chapters are: Introduction to partial differential equations (PDEs) (containing also corresponding physical problems, like heat conduction or thin films), Parabolic equations, Iterative linear algebra (including nonlinear systems, and multigrid, where convergence is proved for the W-cycle), Introduction to finite element methods (FEMs) (starting from examples of physical problems, containing also hierarchical shape functions and quadrature rules for tetrahedra, and general coordinate mappings), proceeding by Finite element theory (here, of course, Sobolev spaces appear, elliptic regularity, also Gårding's inequality, but estimates also for rough inhomogeneities or in the maximum norm), Finite element approximations (containing also a discussion on gaps in the theory developed so far, and nonlinear maps, and a posteriori estimates), Mixed and hybrid finite elements (with saddle point problems, \(H^{\mathrm{div}}\)-, \(H^{\mathrm{curl}} \)-conforming spaces, iterative solution methods for the standard linear saddle point problem), Finite elements for parabolic equations (including convection-diffusion and reaction-diffusion problems), Finite elements and multigrid (here also, along with special features of multigrid for FEM, some material on parabolic equations and on mixed methods), Local refinement (containing also adaptive mesh refinements and mortar methods).NEWLINENEWLINEIn the presentation of the material, the author very carefully handles the assumptions needed whereas proofs are mostly contained only on the CD (in which the book gets above 900 pages, and where constants, formula and theorem counters as well as references provide links to the respective information), and exercises are also added to every section. Finite difference methods are not in the center of the author's interest (``tricky to apply \dots unless the problem domains are rectangular'', p. 78, which is a common though erroneous place, or: ``cannot offer \dots approximations to differential equations with Dirac delta-function forcings'', p. 179 -- however, they can, see the book of Samarskii, Lazarov and Makarov, 1987, in Russian, but there are also corresponding English papers by Raytcho Lazarov, see \url{http://www.math.tamu.edu/~lazarov/}).NEWLINENEWLINEOf course, one could also ask for material on eigenvalues, or on finite volume methods or for more up-to-date references, e.g.\ for convection-diffusion (those given are from 1990--1994), whereas there is the second edition of the basic monograph of Roos, Stynes and Tobiska, from 2008 [\textit{H.-G. Roos} et al., Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. 2nd ed. Berlin: Springer (2008; Zbl 1155.65087)], well in the range of references considered by the author (until about 2009). But the book is so rich that these wishes are of minor importance and would enlarge the book even more. While students like PDFs freely readable in the net, or at most small books, who profits most from studying the present work will be PhD-students and colleagues who prepare for lectures or write papers. For this community, however, the appearance of this book is a gift.NEWLINENEWLINEThe programs enclosed are a special story. On a Microsoft machine (not so on Unix) your first problem may be how to read conveniently the README. And the programs, developed from the freely available deal.II collection, will need editing some files first, and probably you run into problems requiring expert help. Perhaps it would have been more lucky to put them into MATLAB: all universities have it, you are less hampered with details getting old, and MathWork, the owner of MATLAB, would even make advertisement for your book.