This book is about solving partial differential equations (PDEs). Such equa­ tions are used to model a wide range ofphenomena in virtually all fields ofsci­ ence and technology. Inthe 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 in­ tricate features of real life. This change challenges our knowledge in computer science and in numerical analysis. The main objective ofthe present book is to teach modern,advanced tech­ niques for numerical PDE solution. The book also introduces several models arising in fields likefinance, medicine, material technology, and geology. Inor­ der 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 Diff­ pack has support for the advanced numerical methods treated in this book. Most chapters have a part on models and methods, and a part on imple­ mentation and Diffpack programming. The exposition is designed such that readers can focus only on the first part, if desired.

Table of Contents:
1 Parallel Computing.- 1.1 Introduction to Parallel Computing.- 1.2 A Different Performance Model.- 1.3 The First MPI Encounter.- 1.4 Basic Parallel Programming with Diffpack.- 1.5 Parallelizing Explicit FD Schemes.- 1.6 Parallelizing FE Computations on Unstructured Grids.- References.- Overlapping Domain Decomposition Methods.- 2.1 Introduction.- 2.2 The Mathematical Formulations.- 2.3 A 1D Example.- 2.4 Some Important Issues.- 2.5 Components of Overlapping DD Methods.- 2.6 A Generic Implementation Framework.- 2.7 Parallel Overlapping DD Methods.- 2.8 Two Application Examples.- References.- 3 Software Tools for Multigrid Methods.- 3.1 Introduction.- 3.2 Sketch of How Multilevel Methods are Implemented in Diffpack.- 3.3 Implementing Multigrid Methods.- 3.4 Setting up an Input File.- 3.5 Playing Around with Multigrid.- 3.6 Equipping the Poisson2 Solver with Multigrid.- 3.7 Systems of Equations, Linear Elasticity.- 3.8 Nonlinear Problems.- References.- 4 Mixed Finite Elements.- 4.1 Introduction.- 4.2 Model Problems.- 4.3 Mixed Formulation.- 4.4 Some Basic Concepts of a Finite Element.- 4.5 Some Code Examples.- 4.6 Programming with Mixed Finite Elements in a Simulator.- References.- 5 Systems of PDEs and Block Preconditioning.- 5.1 Introduction.- 5.2 Block Preconditioners in General.- 5.3 The Bidomain Equations.- 5.4 Two Saddle Point Problems.- References.- 6 Fully Implicit Methods for Systems of PDEs.- 6.1 Introduction.- 6.2 Implementation of Solvers for PDE Systems in Diffpack.- 6.3 Problem with the Gauss-Seidel Method, by Example.- 6.4 Fully Implicit Implementation.- 6.5 Applications.- 6.6 Conclusion.- References.- 7 Stochastic Partial Differential Equations.- 7.1 Introduction.- 7.2 Some Simple Examples.- 7.3 Solution Methods.- 7.4 Quick Overview of Diffpack Tools.- 7.5Tools for Random Variables.- 7.6 Diffpack Tools for Random Fields.- 7.7 Summary.- 7.A Transformation of Random Variables.- 7.B Implementing a New Distribution.- References.- 8 Using Diffpack from Python Scripts.- 8.1 Introduction.- 8.2 Developing Python Interfaces to C/C++ Functions.- 8.3 Compiling and Linking Wrapper Code with Diffpack.- 8.4 Converting Data between Diffpack and Python.- 8.5 Building an Interface to a More Advanced Simulator.- 8.6 Installing Python, SWIG etc.- 8.7 Concluding Remarks.- References.- 9 Performance Modeling of PDE Solvers.- 9.1 Introduction.- 9.2 Model Problems.- 9.3 Numerical Methods.- 9.4 Total CPU Time Consumption.- 9.5 Solution of Linear Systems.- 9.6 Construction of Linear Systems.- 9.7 Concluding Remarks.- References.- 10 Electrical Activity in the Human Heart.- 10.1 The Basic Physiology.- 10.2 Outline of a Mathematical Model.- 10.3 The Bidomain Model.- 10.4 A Complete Mathematical Model.- 10.5 Physiology of the Heart Muscle Tissue.- 10.6 The Numerical Method.- 10.7 Implementation.- 10.8 Optimization of the Simulator.- 10.9 Simulation Results.- 10.10 Concluding Remarks.- References.- 11 Mathematical Models of Financial Derivatives.- 11.1 Introduction.- 11.2 Basic Assumptions.- 11.3 Forwards and Futures.- 11.4 The Black-Scholes Analysis.- 11.5 European Call and Put Options.- 11.6 American Options.- 11.7 Exotic Options.- 11.8 Hedging.- 11.9 Remarks.- References.- 12 Numerical Methods for Financial Derivatives.- 12.1 Introduction.- 12.2 Model Summary.- 12.3 Monte-Carlo Methods.- 12.4 Lattice Methods.- 12.5 Finite Difference Methods.- 12.6 Finite Element Methods.- References.- 13 Finite Element Modeling of Elastic Structures.- 13.1 Introduction.- 13.2 An Introductory Example; Bar Elements.- 13.3 Another Example; Beam Elements.- 13.4 General Three-Dimensional Elasticity.- 13.5 Degrees of Freedom and Basis Functions.- 13.6 Material Types and Elasticity Matrices.- 13.7 Element Matrices in Local Coordinates.- 13.8 Element Load Vectors in Local Coordinates.- 13.9 Element Matrices and Vectors in Global Coordinates.- 13.10 Element Forces, Stresses, and Strains.- 13.11 Implementation of Structural Elements.- 13.12 Some Example Programs.- 13.13 Test Problems.- 13.14 Summary.- References.- 14 Simulation of Aluminum Extrusion.- 14.1 Introduction.- 14.2 Mathematical Formulation.- 14.3 Finite Element Implementation.- 14.4 Object-Oriented Implementation.- 14.5 Numerical Experiments.- 14.6 Concluding Remarks.- References.- 15 Simulation of Sedimentary Basins.- 15.1 Introduction.- 15.2 The Geomechanical and Mathematical Problem.- 15.3 Numerical Methods.- 15.4 Implementing a Solver for a System of PDEs.- 15.5 Verification.- 15.6 A Magmatic Sill Intrusion Case Study.- 15.7 Concluding Remarks.- References.