Discretization

Abstract

This chapter is devoted to discretization techniques. We start with basic methods for the discretization in time. Besides simple time stepping schemes, we will discuss Galerkin time discretization methods. They have a similar structure to the finite element discretization used in space and they are well suited for adaptivity and optimization problems. After an introduction to the fundamental schemes for parabolic equations, we put the focus on the special needs of temporal discretization methods in fluid mechanics. We continue with the introduction of the finite element methods for spatial discretization. Again, we start by presenting the fundamentals before putting spacial attention to saddle point problems and the nonlinear Navier-Stokes equations. Finally, to prepare the necessary tools for an application to fluid-structure interactions, we discuss interface problems, where the solution of exhibits limited regularity along an internal interface boundary.