In this text, we introduce the basic concepts for the numerical modelling of partial equation differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also others, such as the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws. Furthermore, we provide numerous physical examples which underlie such equations. We then analyze numerical solution methods based on finite elements, finite differences, finite volumes, spectral methods and domain decomposition methods. In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs. It is suitable for students of bachelor and master courses in scientific disciplines, and recommendable to researchers who want to approach this interesting branch of applied mathematics. TOC: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.
