In this informal talk I will review some theoretical and practical aspects related to the finite element approximation of eigenvalue problems arising from PDEs.
The review will cover elliptic eigenvalue problems and eigenvalue problems in mixed form, with particular emphasis on the Maxwell eigenvalue problem.
Other topics can be discussed depending on the interests of the audience, including adaptive schemes, approximation of parametric problems, reduced order models.
 

In this talk we study the numerical approximation of the jump-diffusion McKean--Vlasov SDEs with super-linearly growing drift, diffusion and jump-coefficient. In the first step, we derive the corresponding interacting particle system and define a Milstein-type approximation for this. Making use of the propagation of chaos result and investigating the error of the Milstein-type scheme we provide convergence results for the scheme. In a second step, we discuss potential simplifications of the numerical approximation scheme for the direct approximation of the jump-diffusion McKean--Vlasov SDE. Lastly, we present the results of our numerical simulations.

We propose a density functional theory of Thomas-Fermi-(von Weizsacker) type to describe the response of a single layer of graphene to a charge some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers. We further provide conditions under which those minimizers are unique. The associated Euler-Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. For a class of special potentials, we also establish a precise universal asymptotic decay rate, as well as an exact charge cancellation by the graphene sheet. In addition, we discuss the existence of nodal minimizers which leads to multiple local minimizers in the TFW model. This is a joint work with Cyrill Muratov (University of Pisa).