arXiv subjects

S. Sauter

Publications and source records attributed to S. Sauter.

7 recordsLinked to original sources

arXiv2026

On singular Galerkin discretizations for three models in high-frequency scattering

We consider three common mathematical models for time-harmonic high frequency scattering: the Helmholtz equation in two and three spatial dimensions, a transverse magnetic problem in two dimensions, and Maxwell's equation in three dimensions with dissipative boundary conditions such that the continuous problem is well posed. In this paper, we construct meshes for popular (low order) Galerkin finite element discretizations such that the discrete system matrix becomes singular and the discrete problem is not well posed. This implies that a condition "the finite element space has to be sufficiently rich" in the form of a resolution condition - typically imposed for discrete well-posedness - is not an artifact from the proof by a compact perturbation argument but necessary for discrete stability of the Galerkin discretization.

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arXiv2024

The Green`s function for an acoustic, half-space impedance problem Part II: Analysis of the slowly varying and the plane wave component

We show that the acoustic Green`s function for a half-space impedance problem in arbitrary spatial dimension d can be written as a sum of two terms, each of which is the product of an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.

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arXiv2024

An explicit factorization of the Green's function for an acoustic half-space problem with impedance boundary conditions into an oscillatory exponential and a slowly varying function

In this paper, new representations of the Green's function for an acoustic d-dimensional half-space problem with impedance boundary conditions are presented. The main features of the new representation are: a) in addition to additive terms that appear also in the case of Dirichlet or Neumann boundary conditions, the remaining part of the Green's function is factored into an oscillatory complex exponential function (with the product of the wavenumber and the eikonal as argument) and a remaining function which is slowly varying and hence allows for efficient polynomial approximation; b) the representation is given uniformly for all parameters by a single formula which consists of the product of two analytic functions.

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arXiv2022

The inf-sup constant for $hp$-Crouzeix-Raviart triangular elements

In this paper, we consider the discretization of the two-dimensional stationary Stokes equation by Crouzeix-Raviart elements for the velocity of polynomial order $k\geq1$ on conforming triangulations and discontinuous pressure approximations of order $k-1$. We will bound the inf-sup constant from below independent of the mesh size and show that it depends only logarithmically on $k$. Our assumptions on the mesh are very mild: for odd $k$ we require that the triangulations contain at least one inner vertex while for even $k$ we assume that the triangulations consist of more than a single triangle.

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arXiv2022

Crouzeix-Raviart triangular elements are inf-sup stable

The Crouzeix-Raviart triangular finite elements are $\inf$-$\sup$ stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a {\em conjecture of Crouzeix-Falk} from 1989 for $p=3$. Our proof applies to {\em any odd degree} $p\ge 3$ and hence Crouzeix-Raviart triangular finite elements of degree $p$ in two dimensions and the piecewise polynomials of degree $p-1$ with vanishing integral form a stable Stokes pair {\em for all positive integers} $p$.

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arXiv2021

Critical Functions and Inf-Sup Stability of Crouzeix-Raviart Elements

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order $p\geq5$, $p$ odd, are inf-sup stable for the Stokes problem on triangulations. For $p\geq4$, $p$ even, the stability was proved by Á. Baran and G. Stoyan in 2007 by using the \textit{macroelement technique,} a \textit{dimension formula}, the concept of \textit{critical points} in a triangulation and a representation of the corresponding \textit{critical functions}. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability. The generalization of this theory to cover odd polynomial orders $p\geq5$ is involved; one reason is that the macroelement classes, which have been used for even $p$, are unsuitable for odd $p$. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows.

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arXiv2017

A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L ^\infty -Coefficients

We consider elliptic problems with complicated, discontinuous diffusion tensor $A_{\scriptscriptstyle 0} $. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say $A_{\varepsilon}$, and to use standard finite elements. In \cite{Repin2012} a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error is derived under the assumption that the difference $A_{\scriptscriptstyle 0} -A_{\varepsilon}$ is bounded in the $L^{\infty}$-norm, which requires that the approximation of the coefficient matches the discontinuities of the original coefficient. Therefore this theory is not appropriate for applications with discontinuous coefficients along \textit{complicated, curved} interfaces. Based on bounds for $A_{\scriptscriptstyle 0} -A_{\varepsilon}$ in an $L^{q}$-norm with $q<\infty$ we generalize the combined modelling-discretization strategy to a larger class of coefficients.

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