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!

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!

Reviewer: S.K.Rangarajan (Madras)

##### 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 |