In numerical analysis the design and analysis of computational methods is often based on, and closely linked to, a well-posedness result for the underlying continuous problem. In particular the continuous dependence of the continuous model is inherited by the computational method when such an approach is used. In this talk our aim is to design a stabilised finite element method that can exploit continuous dependence of the underlying physical problem without making use of a standard well-posedness result such as Lax-Milgram's Lemma or The Babuska-Brezzi theorem. This is of particular interest for inverse problems or data assimilation problems which may not enter the framework of the above mentioned well-posedness results, but can nevertheless satisfy some weak continuous dependence properties. First we will discuss non-coercive elliptic and hyperbolic equations where the discrete problem can be ill-posed even for well posed continuous problems and then we will discuss the linear elliptic Cauchy problem as an example of an ill-posed problem where there are continuous dependence results available that are suitable for the framework that we propose.
- Computational Mathematics and Applications Seminar