arXiv subjects

C. Xenophontos

Publications and source records attributed to C. Xenophontos.

4 recordsLinked to original sources

arXiv2026

Analytic regularity for a fourth-order singularly perturbed boundary balue problem with two small parameters

We consider a fourth order singularly perturbed boundary value problem with two small parameters, in one dimension, under the assumption of analytic input data. We show that the solution may be decomposed into a smooth part, two different width boundary layers, and a negligible remainder. We provide estimates for arbitrary order derivatives of each term of the decomposition, which are explicit in the differentiation order and the singular perturbation parameters, and are needed for proving the convergence of high order numerical methods, such as the $p/hp$ versions of the Finite Element Method. We also provide classical differentiability results, which show that the solution will be analytic, if the data are analytic, but negative powers of the singular perturbation parameter(s) show up once we start differentiating.

math.NA↗

arXiv2026

Neural Networks for Singular Perturbations -- Finite Regularity

We study finite-element and deep feedforward neural network (DNN for short) expressivity rate bounds for solution sets of a model linear, second order singularly perturbed, elliptic two-point boundary value problem, in Sobolev norms on a bounded interval $(-1,1)$, with explicit dependence on the singular perturbation parameter $\e\in (0,1]$. Emphasis is on low Sobolev regularity of the data, i.e., source term $f$ and reaction coefficient $b$. A proof of $\e$-explicit solution regularity based on exponentially weighted energy-norm bounds is developed, and \emph{$\e$-robust, algebraic expression rate bounds} in Sobolev norms for $\mathbb{P}_1$ Finite-Elements on exponential and Shishkin type meshes is proved. Expression rates for shallow (fixed depth) $\ReLU$-NNs are shown which are robust w.r. to $\e$ and explicit in terms of the NN size. Robust NN expression rate bounds are further studied for deep feedforward DNNs with ReLU and tanh-activations. As in \cite{OSX24_1085}, tanh- and sigmoid-activated sub-NNs allow to include exponential boundary layer functions exactly into the NN feature space, leading to reduced NN sizes. Recent bitstring encoding techniques for deep NNs with ReLU activations afford, still under low data regularity $f,b \in H^1(I)$ \emph{twice the (robust) convergence rate of $\mathbb{P}_1$ Finite-Elements} achievable with ``eXp'' or Shishkin meshes.

math.NA↗

arXiv2023

Analytic regularity for a singularly perturbed fourth order reaction-diffusion boundary value problem

We consider a fourth order, reaction-diffusion type, singularly perturbed boundary value problem, and the regularity of its solution. Specifically, we provide estimates for arbitrary order derivatves, which are explicit in the singular perturbation parameter as well as the differentiation order. Such estimates are needed for the numerical analysis of high order methods, e.g.hp Finite Element Method (FEM).

math.CA↗

arXiv2020

Mixed $hp$ FEM for singularly perturbed fourth order boundary value problems with two small parameters

We consider fourth order singularly perturbed boundary value problems with two small parameters, and the approximation of their solution by the $hp$ version of the Finite Element Method on the {\emph{Spectral Boundary Layer}} mesh from \cite{MXO}. We use a mixed formulation requiring only $C^{0}$ basis functions in two-dimensional smooth domains. Under the assumption of analytic data, we show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Our theoretical findings are illustrated through numerical examples, including results using a stronger (balanced) norm.

math.NA↗