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
level.