Although I am at ENUMATH since four days, I did not post any news yet.
Most talks here dealt with numerical methods for PDEs and since this is not my primary topic I sometimes had a hard time to grab what the problems and goals were. One exception was the talk by Franco Brezzi. Ha gave an entertaining talk entitled “To reconstruct or not to reconstruct?”. In a nutshell he talked about discretization methods for PDEs by either finite differences and finite elements. He distinguished the two methods by the fact that finite difference methods only use and produce “nodal values”, i.e. values of the solution at specific points. On the other hand, finite element methods also work with a set of values describing the solution. However, these numbers are coefficients to some basis functions and hence, one can “reconstruct” a true function from these values. His first point was, that methods with “reconstruction” usually have much simples proofs for convergence and so on. However, finite difference methods are usually much more simple to implement since the discretization itself explicitly dictates the linear system one has to solve. He then introduced “mimetic finite differences” which, in my incomplete understanding, are a kind of finite difference methods that incorporate more geometric information. During his talk I thought about phrasing his concepts of “reconstruction” and “evaluating” in the language of signal processing as “interpolation” and “sampling” and if this would give another perspective which could be helpful.