Adaptive multilevel solution of nonlinear parabolic PDE systems. Theory, algorithm, and applications.

*(English)*Zbl 0963.65102
Lecture Notes in Computational Science and Engineering. 16. Berlin: Springer. xii, 157 p. (2001).

This book is intended for graduate students and researchers who are interested in the theoretical understanding of systems of nonstationary partial differential equations (PDEs) and who want to use numerical algorithms for their computing. Nonlinear parabolic PDE systems arise in many branches of applications – biology, chemistry, metallurgy, medicine etc.

Adaptive numerical solution of these systems is the main topic of this book from both theoretical and practical point of views. The author follows Rothe’s approach for solving the problem. Convergence results for Rosenbrock methods are sumamarized in this book together with the approximation properties of finite elements to the spatial discretization of the Rosenbrock systems. A posteriori error estimators based on the difference of higher and lower order solutions are also discussed. The spatial error estimator is used to construct an efficient and reliable adaptive strategy for an automatic mesh control.

Illustrative numerical examples that use three Rosenbrock solvers are presented to demonstrate that the theoretical order predictions are indeed of interest for the numerical practice. The last chapter is dedicated to real-life applications that arise in chemical industry, semiconductor-device fabrication and health care.

Adaptive numerical solution of these systems is the main topic of this book from both theoretical and practical point of views. The author follows Rothe’s approach for solving the problem. Convergence results for Rosenbrock methods are sumamarized in this book together with the approximation properties of finite elements to the spatial discretization of the Rosenbrock systems. A posteriori error estimators based on the difference of higher and lower order solutions are also discussed. The spatial error estimator is used to construct an efficient and reliable adaptive strategy for an automatic mesh control.

Illustrative numerical examples that use three Rosenbrock solvers are presented to demonstrate that the theoretical order predictions are indeed of interest for the numerical practice. The last chapter is dedicated to real-life applications that arise in chemical industry, semiconductor-device fabrication and health care.

Reviewer: Angela Handlovičová (Bratislava)

##### MSC:

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

82D55 | Statistical mechanics of superconductors |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |

80A32 | Chemically reacting flows |

80M10 | Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer |