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Sapere aude! Immanuel Kant (1724-1804) Numerical simulations playa key role in many areas of modern science and technology. They are necessary in particular when experiments for the underlying problem are too dangerous, too expensive or not even possible. The latter situation appears for example when relevant length scales are below the observation level. Moreover, numerical simulations are needed to control complex processes and systems. In all these cases the relevant problems may become highly complex. Hence the following issues are of vital importance for a numerical simulation: - Efficiency of the numerical solvers: Efficient and fast numerical schemes are the basis for a simulation of 'real world' problems. This becomes even more important for realtime problems where the runtime of the numerical simulation has to be of the order of the time span required by the simulated process. Without efficient solution methods the simulation of many problems is not feasible. 'Efficient' means here that the overall cost of the numerical scheme remains proportional to the degrees of freedom, i. e. , the numerical approximation is determined in linear time when the problem size grows e. g. to upgrade accuracy. Of course, as soon as the solution of large systems of equations is involved this requirement is very demanding.



1. Wavelet Bases

In this chapter, we describe the general framework of wavelet bases and review some of their particular examples. We will first give the definition and some important features in a rather general setting. This will be useful since this general framework fits to the various applications we will consider later. Then we describe wavelet bases starting on the real line up to general domains.
Karsten Urban

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

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.
Karsten Urban

3. Applications

In this chapter, we describe some applications of the wavelet bases described above. These applications are of two types. First, we use the strong analytical properties combined with the particular construction of wavelets in H(div; Ω) and H(curl; Ω) in order to prove robustness and optimality of wavelet preconditioners in a Wavelet-Galerkin method for certain relevant problems. These problems include the Lamé equations in linear elasticity, Maxwell’s equations in electromagnetism and formulations of the incompressible Navier-Stokes equations in the primitive variables.
Karsten Urban


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