Wavelet Methods for Elliptic Partial Differential Equations

Abstract

Wavelets have become a powerful tool in several applications by now. Their use for the numerical solution of operator equations has been investigated more recently. By now the theoretical understanding of such methods is quite advanced and has brought up deep results and additional understanding. Moreover, the rigorous theoretical foundation of wavelet bases has also lead to new insights in more classical numerical methods for partial differential equations (pde's) such as Finite Elements. However, sometimes it is believed that understanding and applying the full power of wavelets needs a strong mathematical background in functional analysis and approximation theory. The main idea of this book is to introduce the main concepts and results of wavelet methods for solving linear elliptic partial differential equations in a framework that allows avoiding technicalities to a maximum extend. On the other hand, the book also describes recent research including adaptive methods also for nonlinear problems, wavelets on general domains and applications.