## Search

## Partial Differential Equations

Stabilised Finite Element Methods for Variational Inequalities.

*Rolf Stenberg*, Aalto University, Finland.

## Abstract

We survey our recent and ongoing work [1,2] on finite element methods for contact problems. Our approach is to first write the problem in mixed form, in which the contact pressure act as a Lagrange multiplier. In order to avoid the problems related to a direct mixed finite element discretisation, we use a stabilised formulation, in which appropriately weighted residual terms are added to the discrete variational forms. We prove that the formulation is uniformly stable, which implies an optimal a priori error estimate. Using the stability of the continuous problem, we also prove a posteriori estimates, the optimality of which is ensured by local lower bounds. In the implementation of the methods, the discrete Lagrange multiplier is locally eliminated, leading to a Nitsche-type method [3].

For the problems of a membrane and plate subject to solid obstacles, we present numerical results.

Joint work with Tom Gustafsson (Aalto) and Juha Videman (Lisbon).

### References

- T. Gustafsson, R. Stenberg, J. Videman. Mixed and stabilized finite element methods for the obstacle problem. SIAM Journal of Numerical Analysis 55 (2017) 2718–2744
- T. Gustafsson, R. Stenberg, J. Videman. Stabilized methods for the plate obstacle problem. BIT– Numerical Mathematics (2018) DOI: 10.1007/s10543-018-0728-7
- E. Burman, P. Hansbo, M.G. Larson, R. Stenberg. Galerkin least squares finite element method for the obstacle problem. Computer Methods in Applied Mechanics and Engineering 313 (2017) 362–374

## LisMath

An introduction to PT-Symmetric Quantum Theory.

*Paulo Rocha*, LisMath, Faculdade de Ciências.

## Abstract

Traditionally in quantum mechanics it is assumed that the Hamiltonian must be Hermitian in order to obtain real energy levels and unitary time evolution. Here we will show that the requirement of Hermiticity may be replaced by space-time reflection (PT-symmetry) without losing any of the essential physical features of quantum mechanics. In this seminar we will give an introduction to PT-symmetric quantum theory and work with some examples.

## String Theory

Instantons in the Hofstadter butterfly: resurgence and quantum mirror curves.

*Zhihao Duan*, École Normale Supérieure Paris.

## Abstract

Recently an interesting connection between topological string theory and lattice models in condensed matter physics was discussed by several authors. In this talk, we will focus on the Harper-Hofstadter Hamiltonian. For special values of the magnetic flux, its energy spectrum can be exactly solved and its graph has a beautiful shape known as Hofstadter's butterfly. We are interested in the non-perturbative information inside the spectrum. First we consider the weak magnetic field limit and write down a trans-series ansatz for the energies. We then discuss fluctuations around instanton sectors as well as resurgence relations. For the second half of the talk, our goal is to present another powerful way to compute those fluctuations using the topological string formalism, after reviewing all the necessary background. The talk will be based on arXiv: 1806.11092.

## String Theory

Siegel Paramodular Forms and Sparseness in $\operatorname{AdS}_3/\operatorname{CFT}_2$.

*Alexandre Belin*, University of Amsterdam.

## Abstract

We discuss the application of Siegel paramodular forms to the counting of polar states in symmetric product orbifold CFTs.

## String Theory

Counting formulae for extremal black holes in the STU-model.

*Davide Polini*, Instituto Superior Técnico.

## Abstract

We present microstate counting formulae for extremal black holes in the $N=2$ STU-model.