PostDoc Days 2026

We prove optimal convergence rates for certain time discretization schemes applied to the one-dimensional periodic nonlinear Schrödinger and wave equations under the assumption of $H^1$ solutions. For the Schrödinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schrödinger case we use the $L^4$ Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates.

We introduce a data-driven approach to the modelling and analysis of a PDE describing population growth in the stationary case \begin{equation}\label{P} -\Delta u = r u \quad \text{in } \Omega, \quad \frac{\partial u}{\partial\nu}=0 \quad \text{on } \partial\Omega. \end{equation} Instead of considering $(x,u)\mapsto r(x,u)$ as a predefined constitutive law, such as the Verhulst model of logistic growth, we exploit a method based on directly utilizing experimental data $D=\{(x_\beta, u_\beta,r_\beta): \beta\in B\}\subset \mathbb{R}^{n+2}$. Specifically, we look for function pairs $(u,r)\in H^1(\Omega)\times L^\infty(\Omega)$ that satisfy \eqref{P} and match the data in an optimal way. These optimal data-driven solutions provide a new relaxed solution concept, for which we analyze consistency and convergence properties.
The talk is built on work in progress with Wolfgang Reichel (KIT) and Paolo Malanchini (Universitá degli Studi di Milano - Bicocca)

I am interested in periodic waves, which arise across a wide range of disciplines, including biology, ecology, chemistry, and nonlinear optics. The fundamental mechanism describing the emergence of such waves can be traced back to the pioneering work of Alan Turing. Since then, understanding the robustness and stability of these wave patterns have become an active and important research question. In this talk, I will provide a broad overview of the current state of this theory relevant to my research and introduce some theoretical and conceptual ideas.

We are interested in time-periodic, space-localized solutions, called breathers, to semilinear wave equations such as \begin{align*} - \partial_x^2 u + V(x) \partial_t^2 u = \Gamma(x) |u|^{p-1} u = 0 \qquad\text{for}\quad (x, t) \in \mathbb R^2. \end{align*} In order to obtain them one often requires a nonresonance condition, i.e., $0$ is not in the spectrum of the linear operator \begin{align*} -\partial_x^2 + V(x) \partial_t^2 \end{align*} restricted to time-periodic functions, or equivalently, the spectra $\sigma(-\frac{1}{V(x)} \partial_x^2)$ and $\sigma(-\partial_t^2) = \{k^2 \omega^2 \colon k \in \mathbb Z\}$ are separated. For periodic $V$, we develop asymptotics for the spectra of the Sturm-Liouville operator \begin{align*} - \frac{1}{V(x)} \partial_x^2. \end{align*} We show that the nonresonance condition is related to non-smoothness of the potential $V$, and it is achievable (asymptotically) for discontinuous $V$ under mild assumptions. If time allows, we also discuss extensions to perturbations of periodic potentials.
This is based on ongoing work together with Michael Plum.

This talk presents weighted finite difference methods for numerically solving highly oscillatory dispersive PDEs, starting from initial data that are a sum of modulated highly oscillatory exponentials. The proposed methods do not need to resolve high-frequency oscillations in both space and time by prohibitively fine grids as would be required by standard finite difference methods. The approach taken here modifies traditional finite difference methods by appropriate exponential weights. Specifically, we propose weighted leapfrog and weighted Crank—Nicolson methods, both of which achieve second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by the small parameter. Numerical experiments illustrate the theoretical results.

In this talk, we discuss the spectral stability of multi-solitons for the nonlinear Schrödinger (NLS) equation with a periodic potential. These solutions are constructed by concatenating finitely many standing waves. Our results show that their spectral stability is determined by the relative phases of the individual solitons. In particular, we prove that multi-solitons, consisting of mutually out-of-phase 1-solitons, are spectrally stable. The proof relies on the Krein-index count and dynamical systems techniques.
This is joint work with Björn de Rijk.

Since the early days of quantum mechanics, much effort has been devoted to determining the ground-state energies of atoms and molecules as accurately as possible, both experimentally and mathematically. While a large toolbox of methods allows one to derive very precise upper bounds for atomic and molecular ground-state energies, deriving comparable lower bounds is considerably more challenging. In fact, much less rigorous results are known, and many of them still rely on numerical computations in parts of their derivation. In this talk, I will present a new iterative method that enables the computation of rigorous lower bounds for the ground-state energies of all atoms and ions. The accuracy of these lower bounds depends on two ingredients: the quality of the used initial energy estimate and the quality of the used upper bound for the maximum number of bound electrons, $N_c(Z)$, as a function of the nuclear charge $Z>0$. Using this method, we compute lower bounds for the ground-state energies $E(N,N)$ of all atoms with atomic numbers $2 \leq N \leq 86$ and we compare them with previously known lower and upper bounds from the literature. Even more remarkably, our method yields the explicit lower bound $E(N,N) \geq -CN^{7/3}$ for all $N \in \mathbb{N}$ with $C=0.8710$, thereby significantly improving the previously best-known constant $C=(3/2)^{1/3}$. Finally, I will also briefly discuss a joint work with Jannis Anapolitanos, in which we use this method to complete the proof of the van der Waals law for a system of hydrogen and helium atoms.

This talk concerns fully discrete finite element approximations of low-regularity solutions to nonlinear wave equations. We first study semilinear wave equations on bounded polygonal/polyhedral domains with standard boundary conditions, using an exponential Euler scheme combined with conforming finite elements, and establish weak-norm error estimates for initial data in \(H^\gamma(\Omega)\times H^{\gamma-1}(\Omega)\), \(0<\gamma\le 1\). We then extend this low-regularity analysis to the more challenging case of nonlinear wave equations with dynamic boundary conditions on smooth domains, discretized by a fully discrete isoparametric bulk–surface finite element method. In this nonmatching setting, where the continuous and discrete problems are posed on different geometries, we prove weak-norm convergence for finite-energy solutions in \(H^1(\Omega;\Gamma)\times L^2(\Omega;\Gamma)\) by combining frequency truncation with weak-norm lift and adjoint-lift estimates.
These results extend low-regularity error analysis from Fourier and spectral discretizations to finite element methods on general domains, including coupled bulk–surface problems with nontrivial boundary dynamics. In particular, they show that higher-order finite element spaces continue to yield improved spatial convergence rates even for rough solutions. Numerical experiments support the theoretical findings.

In this talk we discuss a well-posedness result of quasilinear Maxwell's equations on the domain $\Omega=(0,T)\times \mathbb{R}^3_+$ with a special type of absorbing boundary conditions given by \[E \times \nu = g-\beta(x)(\nu \times (H \times \nu)). \] Here the resistivity $\beta$ can be seen as a damping term on the boundary, which is allowed to be space-dependent. A wide range of variants of absorbing boundary conditions have been studied in the literature where $\beta$ is uniformly positive definite on the entire domain. In this project, however, we investigate the case where $\beta$ is allowed to vanish on some part of the boundary (e.g. consider two different boundary materials), and the damped and undamped case exist simultaneously, leaving the highly irregular boundary condition $E \times \nu=g$ there.
In the end we obtain well-posedness of the quasi-linear problem for smooth $\beta$. The two main challenges are existence results for appearing traces (which are already an issue for the linear problem on $L^2$), and finding an approximation that takes the compatibility conditions into account to obtain estimates of higher order, which are necessary to close the fixed point argument.