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Introduction to the numerical approximation of Partial Differential Equations
Practical applications Examples from different fields (biology, medicine) and industry
This book aims at introducing students to the numerical approximation of Partial Differential Equations (PDEs). One of the difficulties of this subject is to identify the right trade-off between theoretical concepts and their use in practice. With that collection of examples and exercises we try to address this issue by illustrating "standard" examples which focus on basic concepts of Numerical Analysis, as well as problems derived from practical applications which the student is encouraged to formalize in terms of PDEs, analyze and solve. The latter examples are derived from the experience of the authors in research project developed in collaboration with scientists of different fields (biology, medicine, etc.) and industry. We wanted this book to be useful both to readers more interested in the theoretical aspects, and also to those more concerned with the numerical implementation. To this aim, solutions to the exercises have been subdivided in three parts. The first concerns the mathematical analysis of the problem, the second its numerical approximation and the third part is devoted to implementation aspects and the analysis of the results. The book consists of three parts. The first deals with basic material and results provided as useful reference for the other sections. In particular, we recall the basics of functional analysis and the finite element method. The second part deals with steady elliptic problems, solved with finite elements or finite differences, while in the third part we address time-dependent problems, including linear hyperbolic systems and Navier-Stokes equations. Two appendices discuss some practical implementation issues and three-dimensional applications. Each section contains a brief introduction to the subject to make this book self contained.
Part I Basic Material. 1 Some fundamental tools. 1.1 Hilbert spaces. 1.2 Distributions. 1.3 The spaces Lp and Hs. 1.4 Sequences in lp. 1.5 Important inequalities. 1.6 Brief overview of matrix algebra. 2 Fundamentals of finite elements and finite differences. 2.1 The one dimensional case: approximation by piecewise polynomials. 2.2 Interpolation in higher dimension using finite elements. 2.2.1 Geometric preliminary definitions. 2.2.2 The finite element. 2.2.3 Parametric Finite Elements. 2.2.4 Function approximation by finite elements. 2.3 The method of finite differences. 2.3.1 Difference quotients in dimension one. Part II Stationary Problems. 3 Galerkin-finite element method for elliptic problems. 3.1 Approximation of 1D elliptic problems. 3.1.1 Finite differences in 1D. 3.2 Elliptic problems in 2D. 3.3 Domain decomposition methods for 1D elliptic problems. 3.3.1 Overlapping methods. 3.3.2 Non-overlapping methods. 4 Advection-diffusion-reaction (ADR) problems . 4.1 Preliminary problems. 4.2 Advection dominated problems. 4.3 Reaction dominated problems. Part III Time dependent problems. 5 Equations of parabolic type. 5.1 Finite difference time discretization. 5.2 Finite element time discretization. 6 Equations of hyperbolic type. 6.1 Scalar advection-reaction problems. 6.2 Systems of linear hyperbolic equations of order one. 7 Navier-Stokes equations for incompressible fluids. 7.1 Steady problems. 7.2 Unsteady problems. Part IV Appendices. A The treatment of sparse matrices. A.1 Storing techniques for sparse matrices. A.1.1 The COO format. A.1.2 The skyline format. A.1.3 The CSR format. A.1.4 The CSC format. A.1.5 The MSR format. A.2 Imposing essential boundary conditions. A.2.1 Elimination of essential degrees of freedom. A.2.2 Penalization technique. A.2.3 “Diagonalization” technique. A.2.4 Essential conditions in a vectorial problem. B Who’swho.