Numerical Models for Differential Problems

In this text, we introduce the basic concepts for the numerical modeling 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. We then analyze numerical solution methods based on finite elements, finite differences, finite volumes, spectral methods and domain decomposition methods, and reduced basis methods. 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.

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