Partial Differential Equations: Modelling and Numerical Simulation (Computational Methods in Applied Sciences)

✍ Scribed by Roland Glowinski, Pekka Neittaanmaki

• Publisher: Springer
• Year: 2008
• Tongue: English
• Leaves: 294
• Edition: 1
• Category: Library

✦ Synopsis

This book is dedicated to Olivier Pironneau. For more than 250 years partial differential equations have been clearly the most important tool available to mankind in order to understand a large variety of phenomena, natural at first and then those originating from human activity and technological development. Mechanics, physics and their engineering applications were the first to benefit from the impact of partial differential equations on modeling and design, but a little less than a century ago the Schrödinger equation was the key opening the door to the application of partial differential equations to quantum chemistry, for small atomic and molecular systems at first, but then for systems of fast growing complexity. Mathematical modeling methods based on partial differential equations form an important part of contemporary science and are widely used in engineering and scientific applications. In this book several experts in this field present their latest results and discuss trends in the numerical analysis of partial differential equations. The first part is devoted to discontinuous Galerkin and mixed finite element methods, both methodologies of fast growing popularity. They are applied to a variety of linear and nonlinear problems, including the Stokes problem from fluid mechanics and fully nonlinear elliptic equations of the Monge-Ampère type. Numerical methods for linear and nonlinear hyperbolic problems are discussed in the second part. The third part is concerned with domain decomposition methods, with applications to scattering problems for wave models and to electronic structure computations. The next part is devoted to the numerical simulation of problems in fluid mechanics that involve free surfaces and moving boundaries. The finite difference solution of a problem from spectral geometry has also been included in this part. Inverse problems are known to be efficient models used in geology, medicine, mechanics and many other natural sciences. New results in this field are presented in the fifth part. The final part of the book is addressed to another rapidly developing area in applied mathematics, namely, financial mathematics. The reader will find in this final part of the volume, recent results concerning the simulation of finance related processes modeled by parabolic variational inequalities.

✦ Table of Contents

Contents......Page 10
List of Contributors......Page 12
Part I: Discontinuous Galerkin and Mixed Finite Element Methods......Page 16
Discontinuous Galerkin Methods......Page 17
Mixed Finite Element Methods on Polyhedral Meshes for Diffusion Equations......Page 41
On the Numerical Solution of the Elliptic Monge-Ampère Equation in Dimension Two: A Least-Squares Approach......Page 56
Part II: Linear and Nonlinear Hyperbolic Problems......Page 77
Higher Order Time Stepping for Second Order Hyperbolic Problems and Optimal CFL Conditions......Page 78
Comparison of Two Explicit Time Domain Unstructured Mesh Algorithms for Computational Electromagnetics......Page 105
The von Neumann Triple Point Paradox......Page 123
Part III: Domain Decomposition Methods......Page 139
A Lagrange Multiplier Based Domain Decomposition Method for the Solution of aWave Problem with Discontinuous Coefficients......Page 140
Domain Decomposition and Electronic Structure Computations: A Promising Approach......Page 155
Part IV: Free Surface, Moving Boundaries and Spectral Geometry Problems......Page 173
Numerical Analysis of a Finite Element/Volume Penalty Method......Page 174
A Numerical Method for Fluid Flows with Complex Free Surfaces......Page 193
Modelling and Simulating the Adhesion and Detachment of Chondrocytes in Shear Flow......Page 215
Computing the Eigenvalues of the Laplace-Beltrami Operator on the Surface of a Torus: A Numerical Approach......Page 230
Part V: Inverse Problems......Page 238
A Fixed Domain Approach in Shape Optimization Problems with Neumann Boundary Conditions......Page 239
Reduced-Order Modelling of Dispersion......Page 249
Part VI: Finance (Option Pricing)......Page 261
Calibration of Lévy Processes with American Options......Page 262
An Operator Splitting Method for Pricing American Options......Page 281

✦ Subjects

Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;