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Combined methods for elliptic equations with singularities, interfaces and infinities. (English) Zbl 0909.65079

Mathematics and its Applications (Dordrecht). 444. Dordrecht: Kluwer Academic Publishers. xxiv, 476 p., Dfl 395.00; $ 214.00; £135.00 (1998).
When a computational specialist has to approximate the solution of a partial differential equation, he can choose between different kinds of discretization methods: the finite element method is now widely used but there are many others like the Ritz Galerkin method, the spectral method, the boundary approximation method, the finite difference method, the finite volume method, the boundary element method, the collocation method, the least squares method, and the \(h\)- and \(p\)-methods.
The first objective of this book is to introduce the reader to the bases of these different methods and to discuss for what kind of problems these methods are really performed. In particular, the situation is quite different according the solution to approximate is regular or singular. This is why the author discusses in detail some basic elliptic problems including possible singularities and their treatments. Important applications occur in non-smooth domains or under mixed boundary conditions, like in crack problems.
The second objective of this book is to discuss how to couple two or more of these methods to improve the approximation of a problem which includes, for instance, regular and singular components. Some coupling techniques are thoroughly developed like Lagrange multipliers, penalty techniques, hybrid techniques…
A kind objective of the book is to discuss implementation and stability of numerical solutions, report some numerical experiments, give error bounds and develop a coupling strategy between different approximation methods.
In addition, the author presents in the last part applications and advanced topics: crack-infinity problem, wind flow over buildings, superconvergence, iterative substructuring methods, Schwarz alternating method.
In conclusion, this book presents bases and advanced results on useful combinations between different numerical methods. The writing is clear and attractive, the purpose is well documented so that applied mathematicians, engineers and advanced students should enjoy this monograph.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
74S15 Boundary element methods applied to problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
65N06 Finite difference methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
74R99 Fracture and damage
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
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