This book is a collection of lecture notes for the CIME course on "Multiscale and Adaptivity: Modeling, Numerics and Applications," held in Cetraro (Italy), in July 2009. Complex systems arise in several physical, chemical, and biological processes, in which length and time scales may span several orders of magnitude. Traditionally, scientists have focused on methods that are particularly applicable in only one regime, and knowledge of the system on one scale has been transferred to another scale only indirectly. Even with modern computer power, the complexity of such systems precludes their being treated directly with traditional tools, and new mathematical and computational instruments have had to be developed to tackle such problems. The outstanding and internationally renowned lecturers, coming from different areas of Applied Mathematics, have themselves contributed in an essential way to the development of the theory and techniques that constituted the subjects of the courses.

This book contains an expanded and smoothed version of lecture notes delivered by the authors at the Advanced School on Numerical Solutions of Partial Di?- ential Equations: New Trends and Applications, which took place at the Centre de Recerca Matem atica (CRM) in Bellaterra (Barcelona) from November 15th to 22nd, 2007. The book has three parts. The ?rst part, by Silvia Bertoluzza and Silvia Falletta, is devoted to the use of wavelets to derive some new approaches in the numerical solution of PDEs, showing in particular how the possibility of wr- ing equivalent norms for the scale of Besov spaces allows to write down some new methods.Thesecondpart, byGiovanniRusso, providesanoverviewofthemodern finite-volume and finite-difference shock-capturing schemes for systems of cons- vationandbalancelaws, with emphasisingiving auni?ed viewofsuchschemesby identifying the essential aspects of their construction. In the last part Chi-Wang Shugivesageneralintroductionto thediscontinuousGalerkinmethods forsolving some classes of PDEs, discussing cell entropy inequalities, nonlinear stability and error estimates. The school that originated these notes was born with the objective of p- viding an opportunity for PhD students, recent PhD doctorates and researchers in general in ?elds of applied mathematics and engineering to catch up with - portant developments in the ?elds and/or to get in touch with state-of-the-art numerical techniques that are not covered in usual courses at graduate leve