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Spectral elements for transport-dominated equations. (English) Zbl 0891.65118
Lecture Notes in Computational Science and Engineering. 1. Berlin: Springer. x, 211 p. (1997).
The book deals with the construction of a computational code based on the spectral collocation method using algebraic polynomials. It addresses mainly the elliptic type boundary value problems in 2-D. The emphasis is on the first-order advective terms, in the presence of second-order diffusive terms. Several examples from the various fields of computational physics and engineering are given, to illustrate the methodology and the importance. The chapters are organized as: (1) The Poisson equation in the square (the collocation method, convergence analysis, numerical algorithm, eikonical equation …); (2) Steady transport diffusion equations (the upwind grid, numerical implementation); (3) Other kinds of boundary conditions (Neumann type, boundary conditions in weak form, an approximation of the Poincaré-Steklov operator); (4) The spectral element method (dealing with complex domains, the domain decomposition method); (5) Time discretization (Navier-Stokes, nonlinear Schrödinger and semiconductor device equations); (6) Extensions (a posteriori error estimates, pure hyperbolic, dam problems, 3-D Poisson); and the Appendix outlining basic results needed.
The presentation is lucid, the choice of the subject – matter throughout is very thoughtful, the theory legitimate (and heuristic) and the production of ‘lecture notes’ is of excellent quality. A very welcome entrant to the numerical analysis of differential equations, indeed!

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
76D05 Navier-Stokes equations for incompressible viscous fluids
35J25 Boundary value problems for second-order elliptic equations
76M15 Boundary element methods applied to problems in fluid mechanics
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