Numerical Methods for Elliptic and Parabolic Partial Differential Equations

This graduate-level text provides an application oriented introduction to the numerical methods for elliptic and parabolic partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises. For students with mathematics major it is an excellent introduction to the theory and methods, guiding them in the selection of methods and helping them to understand and pursue finite element programming. For engineering and physics students it provides a general framework for the formulation and analysis of methods. This second edition sees additional chapters on mixed discretization and on generalizing and unifying known approaches; broader applications on systems of diffusion, convection and reaction; enhanced chapters on node-centered finite volume methods and methods of convection-dominated problems, specifically treating the now-popular cell-centered finite volume method; and the consideration of realistic formulations beyond the Poisson's equation for all models and methods.

Auteur

Peter Knabner is Professor emeritus at the University of Erlangen-Nürnberg, where he has led the chair Applied Mathematics I from 1994 to 2020, and also guest professor at the cluster of excellence SimTech of the University of Stuttgart. Knabner'research is focussed on the derivation, analysis and numerical approximation of mathematical models for flow and transport in porous media. with applications in science and technology, in particular in hydrogeology. After the study of Mathematics and Computer Science at the Freie Universität Berlin (diploma in 1972) he earned a PhD from the University of Augsburg in 1983, where he also received a higher doctoral degree (habilitation) in 1988. Peter Knabner is author of more than 180 peer-reviewed publications in applied analysis, numerical mathematics and geohydrology. He is author and co-author of 13 research monographs and textbooks in German and English. Lutz Angermann is Professor of Numerical Mathematics at the Department of Mathematics of the Clausthal University of Technology since 2001. His research is concerned with the development and mathematical analysis of numerical methods for solving partial differential equations with special interests in finite volume and finite element methods and their application to problems in Physics and Engineering. After the study of Mathematics at the State University of Kharkov (now V.N. Karazin Kharkiv National University, Ukraine) he earned a PhD from the University of Technology at Dresden in 1987. The University of Erlangen-Nürnberg awarded him a higher doctoral degree (habilitation) in 1995. From 1998 to 2001, he held the post of an Associate Professor of Numerical Mathematics at the University of Magdeburg. He has authored or co-authored about 100 scientific papers, among them four books as co-author, and he edited two books.

Contenu

For Example: Modelling Processes in Porous Media with Differential Equations.- For the Beginning: The Finite Difference Method for the Poisson Equation.- The Finite Element Method for the Poisson Equation.- The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order.- Grid Generation and A Posteriori Error Estimation.- Iterative Methods for Systems of Linear Equations.- Beyond Coercivity, Consistency and Conformity.- Mixed and Nonconforming Discretization Methods.- The Finite Volume Method.- Discretization Methods for Parabolic Initial Boundary Value Problems.- Discretization Methods for Convection-Dominated Problems.- An Outlook to Nonlinear Partial Differential Equations.- Appendices.