Numerical Methods and Diffpack Programming

## Contents |

The book is suitable for readers with a background in basic finite element and finite difference methods for partial differential equations who wants gentle introductions to advanced topics like parallel computing, multigrid methods, and special methods for systems of PDEs. The goal of all chapters is to 'compute' solutions to problems, hence algorithmic and software issues play a central role. All software examples use the Diffpack programming environment, so to take advantage of these examples some experience with Diffpack is required. There are also some chapters covering complete applications, i.e., the way from a model, expressed as systems of PDEs, through discretization methods, algorithms, software design, verification, and computational examples.

This book is about solving partial differential equations (PDEs). Such
equations are used to model a wide range of phenomena in virtually all
fields of science and technology. In the last decade, the general
availability of extremely powerful computers has shifted the focus in
computational mathematics from simplified model problems to much more
sophisticated models resembling intricate features of real life. This
change challenges our knowledge in computer science and in numerical
analysis.

The main objective of the present book is to teach modern, advanced
techniques for numerical PDE solution. The book also introduces several
models arising in fields like finance, medicine, material technology,
and geology. In order to read this book, you must have a basic knowledge
of partial differential equations and numerical methods for solving such
equations. Furthermore, some background in finite element methods is
required. You do not need to know Diffpack, although this programming
environment is used in examples throughout the text. Basically, this
book is about models, methods, and how to implement the methods. For the
implementation part it is natural for us to use Diffpack as the
programming environment, because making a PDE solver in Diffpack
requires little amount of programming and because Diffpack has support
for the advanced numerical methods treated in this book. Most chapters
have a part on models and methods, and a part on implementation and
Diffpack programming. The exposition is designed such that readers can
focus only on the first part, if desired. The prerequisites for the
present book (PDEs and numerics) as well as an introduction to Diffpack
programming can be found in "Computational Partial Differential
Equations - Numerical Methods and Diffpack Programming" by H. P.
Langtangen.

- X. Cai, E. Acklam, H. P. Langtangen, A. Tveito: Parallel Computing.
- X. Cai: Overlapping Domain Decomposition Methods.
- K.-A. Mardal, G. W. Zumbusch, H. P. Langtangen: Software Tools for Multigrid Methods.
- K.-A. Mardal, H. P. Langtangen: Mixed Finite Elements.
- K.-A. Mardal, J. Sundnes, H. P. Langtangen, A. Tveito: Systems of PDEs and Block Preconditioning.
- Å Ødegård, H. P. Langtangen, A. Tveito: Object-Oriented Implementation of Fully Implicit Methods for Systems of PDEs.
- H. P. Langtangen, H. Osnes: Stochastic Partial Differential Equations.
- H. P. Langtangen and K.-A. Mardal: Using Diffpack from Python Scripts.
- X. Cai, A. M. Bruaset, H. P. Langtangen, G. T. Lines, K. Samuelsson, W. Shen, A. Tveito, G. Zumbusch: Performance Modeling of PDE Solvers.
- J. Sundnes, G.T. Lines, P. Grøttum, A. Tveito: Numerical Methods and Software for Modeling the Electrical Activity in the Human Heart.
- O. Skavhaug, B. F. Nielsen, A. Tveito: Mathematical Models of Financial Derivatives.
- O. Skavhaug, B. F. Nielsen, A. Tveito: Numerical Methods for Financial Derivatives.
- T. Thorvaldsen, H. P. Langtangen, H. Osnes: Finite Element Modeling of Elastic Structures.
- K. M. Okstad, T. Kvamsdal: Simulation of Aluminum Extrusion.
- A. Kjeldstad, H. P. Langtangen, J. Skogseid, K. Bjørlykke: Simulation of Deformations, Fluid Flow and Heat Transfer in Sedimentary Basins

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