arXiv subjects

David Wörgötter

Publications and source records attributed to David Wörgötter.

4 recordsLinked to original sources

arXiv2026

Wavenumber-explicit $hp$-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media

We consider the time-harmonic Maxwell equations with impedance boundary conditions on a bounded Lipschitz domain $Ω$ with analytic boundary $Γ$. We suppose that $Ω$ consists of multiple subdomains, and that the permeability and permittivity tensors are analytic on every subdomain, but may jump across subdomain interfaces. Under these conditions we show that for any wavenumber $k\in\mathbb{C}$ with $|k|\geq 1$ for which Maxwell's equations are polynomially well-posed, a Galerkin discretization based on Nédélec elements of order $p$ on a mesh with mesh width $h$ is quasi-optimal, provided that there holds the wavenumber-explicit scale resolution condition a) that $|k|h/p$ is sufficiently small and b) that $p/\log |k|$ is bounded from below.

math.NA↗

arXiv2026

Wavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations at a nonzero wavenumber $k\in\mathbb{C}$ on a bounded and simply connected Lipschitz domain $Ω$ with an analytic boundary $Γ$, on which we impose impedance boundary conditions. We suppose that the (possibly complex-valued) permeability and permittivity tensor fields $\boldsymbolμ^{-1}$ and $\boldsymbol{\varepsilon}$ are piecewise analytic in $Ω$ and discontinuous only across certain mutually disjoint analytic surfaces inside of $Ω$. We show that under these circumstances, any weak solution of Maxwell's equations is piecewise analytic in $Ω$ and that the growth of its derivatives can be controlled explicitly in the wavenumber $k$.

math.AP↗

arXiv2025

Regularity of vector fields with piecewise regular curl and divergence

We consider a bounded Lipschitz domain $Ω\subseteq\mathbb{R}^3$ with sufficiently smooth boundary and prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may be discontinuous across mutually disjoint and sufficiently smooth surfaces inside of $Ω$. The main idea behind our approach is to employ recently developed parametrices for the curl-operator and the regularity theory of Poisson transmission problems. We conclude our work by applying our findings to the heterogeneous time-harmonic Maxwell equations with either a) impedance, b) natural or c) essential boundary conditions and providing wavenumber-explicit piecewise regularity estimates for these equations.

math.AP↗

arXiv2022

Two-level error estimation for the integral fractional Laplacian

For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement.

math.NA↗