##
**Multilevel adaptive methods for partial differential equations.**
*(English)*
Zbl 0707.65080

Frontiers in Applied Mathematics, 6. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. ix, 162 p. $ 24.50 (1989).

The book is written as a practical handbook for efficient numerical solution of partial differential equations. It covers the process from derivation of discretization based on the so-called finite volume element method till the efficient parallel implementation of solution methods for the discrete systems of linear algebraic equations.

The main feature of the book is the development of efficient solution methods based on a sequence of discretizations which by nature are derived on locally refined grid in order to take advantage of possible local properties of the solution of the underlined differential problem. The solution methods are constructed on the basis of the multigrid technique however in the context of local refinement.

There is an introductory chapter for multigrid methods. The ultimate requirement in this process is that the whole (composite) grid to be a union of uniform pieces (patches). In this way the solvers constructed can exploit nice vectorizable and parallelizable properties on such uniformly refined patches. There are two chapters in the book to these methods: The FAC (fast adaptive composite grid) method proposed by the author [Defect correction methods, Theory and Applications, Comput. Suppl. 5, 115–121 (1984; Zbl 0552.65071)] and a parallel version of it - the so-called AFAC (asynchronous FAC) method for efficient parallel implementation of the FAC.

The methods in the book are well illustrated by many numerical examples. It should be pointed out that in this area there is still an on-going research, especially in the multilevel adaptive methods. The book gives, however a current state of the art in this field.

The main feature of the book is the development of efficient solution methods based on a sequence of discretizations which by nature are derived on locally refined grid in order to take advantage of possible local properties of the solution of the underlined differential problem. The solution methods are constructed on the basis of the multigrid technique however in the context of local refinement.

There is an introductory chapter for multigrid methods. The ultimate requirement in this process is that the whole (composite) grid to be a union of uniform pieces (patches). In this way the solvers constructed can exploit nice vectorizable and parallelizable properties on such uniformly refined patches. There are two chapters in the book to these methods: The FAC (fast adaptive composite grid) method proposed by the author [Defect correction methods, Theory and Applications, Comput. Suppl. 5, 115–121 (1984; Zbl 0552.65071)] and a parallel version of it - the so-called AFAC (asynchronous FAC) method for efficient parallel implementation of the FAC.

The methods in the book are well illustrated by many numerical examples. It should be pointed out that in this area there is still an on-going research, especially in the multilevel adaptive methods. The book gives, however a current state of the art in this field.

Reviewer: Panayot S. Vassilevski (Sofia)

### MSC:

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65F10 | Iterative numerical methods for linear systems |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |