Christos Xenophontos

Jens Markus Melenk, Christos Xenophontos, Lisa Oberbroeckling

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< ε\le μ\le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $ε$ and $% μ$. We construct full asymptotic expansions together with error bounds that cover the complete range $0 < ε\leq μ\leq 1$. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.

Sebastian Franz, Christos Xenophontos

In this short note we analyse a connection between the exponentially graded and the class of S-type meshes for singularly perturbed problems. As a by-product we obtain a slightly modified and more general class of layer-adapted meshes.

Joost A. A. Opschoor, Christoph Schwab, Christos Xenophontos

We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval $(-1,1)$. We assume that the given source term and reaction coefficient are analytic in $[-1,1]$. We establish expression rate bounds in Sobolev norms in terms of the NN size which are uniform with respect to the singular perturbation parameter for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and $\tanh$- and sigmoid-activated NNs. The latter activations can represent ``exponential boundary layer solution features'' explicitly, in the last hidden layer of the DNN, i.e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. We prove that all DNN architectures allow robust exponential solution expression in so-called `energy' as well as in `balanced' Sobolev norms, for analytic input data.

Jens Markus Melenk, Christos Xenophontos

The $hp$-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed in one- and two-dimensional domains with analytic boundary. On suitably designed \emph{Spectral Boundary Layer meshes}, robust exponential convergence in a balanced norm is shown. This balanced norm is stronger than the energy norm in that the boundary layers are $O(1)$ uniformly in the singular perturbation parameter. Robust exponential convergence in the maximum norm is also established. The theoretical findings are illustrated with two numerical experiments.