Adaptive Multilevel Solution of Nonlinear Parabolic Pde Systems

by Lang, Jens

• ISBN13: 9783540679004
• ISBN10: 3540679006
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Springer Verlag

Summary

This book deals with the adaptive numerical solution of parabolic partial differential equations (PDEs) arising in many branches of applications. It illustrates the interlocking of numerical analysis, the design of an algorithm and the solution of practical problems. In particular, a combination of Rosenbrock-type one-step methods and multilevel finite elements is analysed. Implementation and efficiency issues are discussed. Special emphasis is put on the solution of real-life applications that arise in today's chemical industry, semiconductor-device fabrication and health care. The book is intended for graduate students and researchers who are either interested in the theoretical understanding of instationary PDE solvers or who want to develop computer codes for solving complex PDEs.

Table of Contents

Preface	vii
Acknowledgements	ix
Introduction	1	(4)
The Continuous Problem and Its Discretization in Time	5	(10)
Nonlinear Evolution Problem	5	(6)
Rosenbrock Methods and Basic Results	11	(4)
Convergence of the Discretization in Time and space	15	(16)
Finite Element Discretization in Space	16	(4)
Proof of the Convergence Results	20	(11)
Computational Error Estimation	31	(16)
Control of Time Steps	31	(4)
Estimation of Spatial Errors	35	(8)
Proof of the Error Estimates	43	(4)
Towards an Effective Algorithm. Practical Issues	47	(22)
Implementation of Rosenbrock Methods	48	(8)
Selected Rosenbrock Methods	56	(2)
Implementation of Multilevel Finite Element Methods	58	(7)
Construction of Improved Spatial Grids	58	(4)
Temporal Grid Evolution	62	(3)
Linear System Solvers	65	(1)
KARDOS - an Accurate Adaptive PDE Solver	65	(4)
Illustrative Numerical Tests	69	(10)
Practical Convergence Observations	69	(6)
Accuracy of the Spatial A Posteriori Error Estimate	75	(1)
Performance of the Multilevel Strategy	76	(3)
Application from Computational Sciences	79	(40)
Two-Phase Bubble Reactor	80	(3)
Propagation of Laminar Flames	83	(9)
Laminar Flames Through an Obstacle	83	(6)
Reaction Front in a Solid	89	(3)
Dopant Diffusion in Silicon	92	(13)
Diffusion Model under Extrinsic Conditions	93	(7)
Some Simulation Results for Phosphorus Diffusion	100	(5)
Bio-Heat Transfer in Regional Hyperthermia	105	(14)
Mathematical Modeling and Optimization	107	(6)
Simulation for Two Individual Patients	113	(6)
Appendix A. Advanced Tools from Functional Analysis	119	(8)
1. Gelfand Triple	119	(1)
2. Sesquilinear Forms and Bounded Operators in Hilbert Spaces	120	(2)
3. Unbounded Operators in Hilbert Spaces	122	(1)
4. Analytic Semigroups	122	(2)
5. Vectorial Functions Defined on Real Intervals	124	(3)
Appendix B. Consistency and Stability of Rosenbrock Methods	127	(4)
1. Order Conditions	127	(1)
2. The Stability Function	127	(1)
3. The Property 'Stiffly Accurate'	128	(3)
Appendix C. Coefficients of Selected Rosenbrock Methods	131	(8)
Appendix D. Color Plates	139	(4)
Table of Notations	143	(2)
Bibliography	145	(10)
Index	155