Wavelet Bases for H(div) and H(curl)

Abstract

In this chapter, we follow mostly [118] and construct wavelet bases specifically for the spaces H(div; Ω) and H(curl; Ω). These spaces arise naturally in the variational formulation of a whole variety of partial differential equations. Two prominent examples are the Navier-Stokes equations that describe the flow of a viscous, incompressible fluid and Maxwell’s equations in electromagnetism. For the incompressible Navier-Stokes equations, H(div; Ω) plays an important role for modeling the velocity-field of the flow. The space H(curl; Ω) has to be considered, when one is interested in a formulation in non-primitive variables such as stream function, vorticity and vector potential, [77]. Certain electromagnetic phenomena are known to be modeled by Maxwell’s equations. Here, the space H(curl; Ω) appears when linking the quantities electric and magnetic field, magnetic induction and flux density, see for example [10, 17, 86], the references therein and also Chap. 3 on p. 109 below.