Foundations of adaptive finite element methods for PDEs

Project description

Foundations of adaptive finite element methods for PDEs (Or- Why do adaptive methods work so well?) 

Adaptive finite element methods allow the computation of solutions to partial differential equations (PDEs) in the most optimal manner that is possible. In particular, these methods require the least amount of degrees-of-freedom to obtain a solution up to a desired accuracy! In recent years a theory has emerged that explains this behaviour. It relies on classical a posteriori error estimation, Banach contraction, and nonlinear approximation theory. Unfortunately, the theory so far applies only to specific model problems.

Challenges for students:

How can the theory be extended to, for example, nonsymmetric problems, nonlinear problems, or time-dependent problems?

What about formulations in Banach spaces instead of the usual Hilbert spaces?

Depending on the interest of the student, several of these issues (or others) can be addressed. Also, the student is encouraged to suggest a second supervisor, possibly from another group!