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The construction of wavelets on general domains is performed in three steps. This chapter starts by introducing the construction of scaling functions and wavelets on bounded univariate intervals. Next, building tensor products allows constructing wavelets on rectangular domains. Finally, the Wavelet Element Method (WEM) is introduced. Using non-overlapping domain decomposition and mapping to the unit cube the WEM matches scaling functions and wavelets across the interfaces of the subdomains in order to obtain a globally continuous basis. The realization of the construction in terms of software is shown as well how to use this software.

This chapter starts by adaptively approximating a given function and introduce the main theoretical concepts. After describing the more classical approach to adaptivity based upon a-posteriori error estimates, it shows the somewhat different perspective of adaptive wavelet methods. Given an operator equation, an equivalent infinite-dimensional problem on sequence spaces is introduced. In order to obtain a computable method, the infinite operator is replaced by an approximate application. The resulting schemes are described, analyzed and compared by numerical experiments. An outlook to nonlinear operators is given.

Chapter 9 studies S. Dahlke studies wavelet methods. This is the first presentation in this generality and is appropriate for nonlinear problems and bifurcation for elliptic PDEs. As a consequence of the difficulties with evaluating nonlinear functionals and operators with wavelet arguments, general quasilinear, and fully nonlinear problems are limited, and excluded, respectively. With this exception, the whole spectrum of corresponding wavelet methods is shown to be stable and convergent. Again the corresponding linearized operator has to be boundedly invertible. This chapter finishes with adaptive wavelet methods. In contrast to Chapter 6, nonlinear approximation, and wavelet matrix compression are employed for adaptive descent iterations.

Some recent applications of wavelet methods are shown. The chapter starts with the numerical realization of the WEM and consider the L-shaped domain as a model for a non-separable domain. The chapter tests adaptive wavelet schemes. Next, more complicated domains are considered. In particular, the role of the mapping and matching approach is investigated. Saddle point problems can be seen as a system of equations that are indefinite. It is shown that an adaptive scheme makes usual compatibility constraints void (Ladyshenskaja-Babushka-Brezzi condition). As a particular example, the chapter considers the Stokes problem from Fluid Dynamics.