In this research topic convergence properties of finite element methods and discontinuous Galerkin methods are analyzed.

I investigate space fractional diffusion probelms bith from the modeling and the analysis point of view. This also includes the numerical solution: both the construction of efficient numericalmethods and their error analysis. 

Neural networks can also be used to the efficient numerical solution of ODE's or PDE's. The chief question is this research topic is the construction of an adequate network with appropriate training data and loss function.