##
**The meshless local Petrov-Galerkin (MLPG) method.**
*(English)*
Zbl 1012.65116

Encino, CA: Tech Science Press. viii, 429 p. (2002).

As the title suggests, the book is concerned with the meshless local Petrov-Galerkin (MLPG) method. It contains a detailed introduction into the MLPG method, as well as two thorough expositions on applications of the MLPG method and its variations on boundary value problems arising in Solid Mechanics and Fluid Mechanics. Finally, an outlook on future research is included. The book features many descriptions of ongoing research activities in various scientific areas relevant for the developed topic. Many useful hints and references are given, making it an ideal basis for further research.

Within the first chapter, basic mathematical formulations concerning the weak formulations of boundary value problems are given. Various aspects of well-known numerical methods as e.g. the Galerkin finite element method or the boundary element method are discussed with respect to their applicability in a meshless framework. Based on the second chapter about meshless interpolation of trial and test functions, the MLPG method and some of its variations are developed within the third chapter. After the discussion of various numerical methods in the first chapter, it especially becomes evident that the main property of the MLPG method is that no domain or boundary mesh is required, i.e. the method is truly meshless. The fourth and fifth chapter contain various applications of the described method in Solid Mechanics and Fluid Mechanics, respectively. Discussed are elasticity problems and steady-state flows.

The MLPG method enables the construction of a locking-free algorithm, and it is also shown that upwinding techniques can be incorporated to enable the treatment of convection dominated problems. The book finishes with an exposition on prospects for future research.

Within the first chapter, basic mathematical formulations concerning the weak formulations of boundary value problems are given. Various aspects of well-known numerical methods as e.g. the Galerkin finite element method or the boundary element method are discussed with respect to their applicability in a meshless framework. Based on the second chapter about meshless interpolation of trial and test functions, the MLPG method and some of its variations are developed within the third chapter. After the discussion of various numerical methods in the first chapter, it especially becomes evident that the main property of the MLPG method is that no domain or boundary mesh is required, i.e. the method is truly meshless. The fourth and fifth chapter contain various applications of the described method in Solid Mechanics and Fluid Mechanics, respectively. Discussed are elasticity problems and steady-state flows.

The MLPG method enables the construction of a locking-free algorithm, and it is also shown that upwinding techniques can be incorporated to enable the treatment of convection dominated problems. The book finishes with an exposition on prospects for future research.

Reviewer: Michael Breuß (Hamburg)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

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

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

74B05 | Classical linear elasticity |

65N38 | Boundary element 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 |

76M10 | Finite element methods applied to problems in fluid mechanics |

76M15 | Boundary element methods applied to problems in fluid mechanics |