In this text, we introduce the basic concepts for the numerical modelling of partial differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations. In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs. The text does not require any…mehr
In this text, we introduce the basic concepts for the numerical modelling of partial differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations. In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs. The text does not require any previous advanced mathematical knowledge of partial differential equations: the absolutely essential concepts are reported in a preliminary chapter. It is therefore suitable for students of bachelor and master courses in scientific disciplines, and recommendable to those researchers in the academic and extra-academic domain who want to approach this interesting branch of applied mathematics.
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Autorenporträt
The Author is Professor and Director of the Chair of Modelling and Scientific Computing (CMCS) at the Institute of Analysis and Scientific Computing of EPFL, Lausanne (Switzerland), since 1998, Professor of Numerical Analysis at the Politecnico di Milano (Italy) since 1989, and Scientific Director of MOX, since 2002. Author of 22 books published with Springer, and of about 200 papers published in refereed international Journals, Conference Proceedings and Magazines, Alfio Quarteroni is actually one of the strongest and reliable mathematicians in the world in the field of Modelling and SC.
Inhaltsangabe
Introduction.- 1 A brief survey on partial differential equations.- 2 Elliptic equations.- 3 The Galerkin finite element method for elliptic problems.- 4 Spectral methods.- 5 Diffusion-transport-reaction equations.- 6 Parabolic equations.- 7 Finite differences for hyperbolic equations.- 8 Finite elements and spectral methods for hyperbolic equations.- 9 Nonlinear hyperbolic problems.- 10 The Navier-Stokes equations.- 11 Finite element programming.- 12 Generation of 1D and 2D grids.- 13 The finite volume method.- 14 Domain decomposition method.- 15 Optimal control problems for partial differential equations.- 16 Reduced basis methods.- 17 Appendix A: Elements of functional analysis.- 18 Appendix B: Solution of algebraic systems.
1 A brief survey of partial differential equations.- 2 Elements of functional analysis.- 3 Elliptic equations.- 4 The Galerkin finite element method for elliptic problems.- 5 Parabolic equations.- 6 Generation of 1D and 2D grids.- 7 Algorithms for the solution of linear systems.- 8 Elements of finite element programming.- 9 The finite volume method.- 10 Spectral methods.- 11 Isogeometric analysis.- 12 Discontinuous element methods (D Gandmortar).- 13 Diffusion-transport-reaction equations.- 14 Finite differences for hyperbolic equations.- 15 Finite elements and spectral methods for hyperbolic equations.- 16 Nonlinear hyperbolic problems.- 17 Navier-Stokes equations.- 18 Optimal control of partial differential equations.- 19 Domain decomposition methods.- 20 Reduced basis approximation for parametrized partial differential equations.- References
Introduction.- 1 A brief survey on partial differential equations.- 2 Elliptic equations.- 3 The Galerkin finite element method for elliptic problems.- 4 Spectral methods.- 5 Diffusion-transport-reaction equations.- 6 Parabolic equations.- 7 Finite differences for hyperbolic equations.- 8 Finite elements and spectral methods for hyperbolic equations.- 9 Nonlinear hyperbolic problems.- 10 The Navier-Stokes equations.- 11 Finite element programming.- 12 Generation of 1D and 2D grids.- 13 The finite volume method.- 14 Domain decomposition method.- 15 Optimal control problems for partial differential equations.- 16 Reduced basis methods.- 17 Appendix A: Elements of functional analysis.- 18 Appendix B: Solution of algebraic systems.
1 A brief survey of partial differential equations.- 2 Elements of functional analysis.- 3 Elliptic equations.- 4 The Galerkin finite element method for elliptic problems.- 5 Parabolic equations.- 6 Generation of 1D and 2D grids.- 7 Algorithms for the solution of linear systems.- 8 Elements of finite element programming.- 9 The finite volume method.- 10 Spectral methods.- 11 Isogeometric analysis.- 12 Discontinuous element methods (D Gandmortar).- 13 Diffusion-transport-reaction equations.- 14 Finite differences for hyperbolic equations.- 15 Finite elements and spectral methods for hyperbolic equations.- 16 Nonlinear hyperbolic problems.- 17 Navier-Stokes equations.- 18 Optimal control of partial differential equations.- 19 Domain decomposition methods.- 20 Reduced basis approximation for parametrized partial differential equations.- References
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