Focusing on wavelet methods for partial differential equations, this text serves as an introduction for graduates, detailing the construction and analysis of adaptive wavelet techniques. It is based on the author's course, providing a comprehensive foundation in the subject, making it a valuable resource for those looking to understand and apply these mathematical methods effectively.
An Introduction to Analytical and Numerical Methods
The textbook offers a comprehensive introduction to partial differential equations through a blend of analytical and numerical methods. By combining these complementary approaches, it establishes a solid foundation for advanced study. The inclusion of motivating examples from physical sciences, engineering, and economics enriches the learning experience, making complex concepts more accessible and relevant.
Problem Adapted Construction and Applications
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.