arXiv subjects

Stefan A. Sauter

Publications and source records attributed to Stefan A. Sauter.

14 recordsLinked to original sources

arXiv2026

Dirichlet-to-Neumann operator for the Helmholtz problem with general wavenumbers on the $n$-sphere

This paper considers the Helmholtz problem in the exterior of a ball with Dirichlet boundary conditions and radiation conditions imposed at infinity. The differential Helmholtz operator depends on the complex wavenumber with non-negative real part and is formulated for general spatial dimensions. We prove wavenumber explicit continuity estimates of the corresponding Dirichlet-to-Neumann (DtN) operator which do not deteriorate as the complex wavenumber tends to zero. The exterior Helmholtz problem can be equivalently reformulated on a bounded domain with DtN boundary conditions on the artificial boundary of a ball. We derive wavenumber independent trace and Friedrichs-type inequalities for the solution space in wavenumber-indexed norms.

math.AP↗

arXiv2026

BEM for variable coefficient second-order problems

A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a computable approximation of the boundary operator from a Galerkin discretisation of the underlying elliptic differential operator in a one-time preprocessing step, for instance by conforming finite elements. The resulting algebraic formulation retains the dimension reduction intrinsic to boundary integral methods and is compatible with standard data-sparse matrix compression techniques.

math.NA↗

arXiv2026

Discrete reliability for high-order Crouzeix--Raviart finite elements

In this paper, the adaptive numerical solution of a 2D Poisson model problem by Crouzeix-Raviart elements ($\operatorname*{CR}_{k}$ $\operatorname*{FEM}$) of arbitrary odd degree $k\geq1$ is investigated. The analysis is based on an established, abstract theoretical framework: the \textit{axioms of adaptivity} imply optimal convergence rates for the adaptive algorithm induced by a residual-type a posteriori error estimator. Here, we introduce the error estimator for the $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ discretization and our main theoretical result is the proof ot Axiom 3: \textit{discrete reliability}. This generalizes results for adaptive lowest order $\operatorname*{CR}_{1}$ $\operatorname*{FEM}$ in the literature. For this analysis, we introduce and analyze new local quasi-interpolation operators for $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ which are key for our proof of discrete reliability. We present the results of numerical experiments for the adaptive version of $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ for some low and higher (odd) degrees $k\geq1$ which illustrate the optimal convergence rates for all considered values of $k$.

math.NA↗

arXiv2024

Explicit Local Time-Stepping for the Inhomogeneous Wave Equation with Optimal Convergence

Adaptivity and local mesh refinement are crucial for the efficient numerical simulation of wave phenomena in complex geometry. Local mesh refinement, however, can impose a tiny time-step across the entire computational domain when using explicit time integration. By taking smaller time-steps yet only inside locally refined regions, local time-stepping methods overcome the stringent CFL stability restriction imposed on the global time-step by a small fraction of the elements without sacrificing explicitness. In [21], a leapfrog based local time-stepping method was proposed for the inhomogeneous wave equation, which applies standard leapfrog time-marching with a smaller time-step inside the refined region. Here, to remove potential instability at certain time-steps, a stabilized version is proposed which leads to optimal L2-error estimates under a CFL condition independent of the coarse-to-fine mesh ratio. Moreover, a weighted transition is introduced to restore optimal H1-convergence when the source is nonzero across the coarse-to-fine mesh interface. Numerical experiments corroborate the theoretical error estimates and illustrate the usefulness of these improvements.

math.NA↗

arXiv2024

The pressure-wired Stokes element: a mesh-robust version of the Scott-Vogelius element

The Scott-Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order $k$ and a discontinuous pressure approximation of order $k-1$. It employs a "singular distance" (measured by some geometric mesh quantity $ Θ\left( \mathbf{z}\right) \geq 0$ for triangle vertices $\mathbf{z}$) and imposes a local side condition on the pressure space associated to vertices $\mathbf{z}$ with $Θ\left( \mathbf{z}\right) =0$. The method is inf-sup stable for any fixed regular triangulation and $k\geq 4$. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices $0<Θ\left( \mathbf{z}\right) \ll 1$. In this paper, we introduce a very simple parameter-dependent modification of the Scott-Vogelius element such that the inf-sup constant is independent of nearly-singular vertices. We will show by analysis and also by numerical experiments that the effect on the divergence-free condition for the discrete velocity is negligibly small.

math.NA↗

arXiv2024

Pressure-improved Scott-Vogelius type elements

The Scott-Vogelius element is a popular finite element for the discretization of the Stokes equations which enjoys inf-sup stability and gives divergence-free velocity approximation. However, it is well known that the convergence rates for the discrete pressure deteriorate in the presence of certain $critical$ $vertices$ in a triangulation of the domain. Modifications of the Scott-Vogelius element such as the recently introduced pressure-wired Stokes element also suffer from this effect. In this paper we introduce a simple modification strategy for these pressure spaces that preserves the inf-sup stability while the pressure converges at an optimal rate.

math.NA↗

arXiv2024

Skeleton Integral Equations for Acoustic Transmission Problems with Varying Coefficients

In this paper we will derive an non-local (``integral'') equation which transforms a three-dimensional acoustic transmission problem with \emph{variable} coefficients, non-zero absorption, and mixed boundary conditions to a non-local equation on a ``skeleton'' of the domain $Ω\subset\mathbb{R}^{3}$, where ``skeleton'' stands for the union of the interfaces and boundaries of a Lipschitz partition of $Ω$. To that end, we introduce and analyze abstract layer potentials as solutions of auxiliary coercive full space variational problems and derive jump conditions across domain interfaces. This allows us to formulate the non-local skeleton equation as a \emph{direct method} for the unknown Cauchy data of the solution of the original partial differential equation. We establish coercivity and continuity of the variational form of the skeleton equation based on auxiliary full space variational problems. Explicit expressions for Green's functions is not required and all our estimates are \emph{explicit} in the complex wave number.

math.AP↗

arXiv2023

Wavenumber-explicit hp-FEM analysis for Maxwell's equations with impedance boundary conditions

The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in k and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nedelec elements of order p on a mesh with mesh size h is shown under the k-explicit scale resolution condition that a) kh/p is sufficient small and b) p/\ln k is bounded from below.

math.NA↗

arXiv2020

A Stable Boundary Integral Formulation of an Acoustic Wave Transmission Problem with Mixed Boundary Conditions

In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. The transmission interfaces may join the domain boundary in a general way independent of the location of the boundary conditions. We will derive a formulation as a \textit{direct}, \textit{space-time retarded boundary integral equation}, where both Cauchy data are kept as unknowns on the impedance part of the boundary. This requires the definition of single-trace spaces which incorporate homogeneous Dirichlet and Neumann conditions on the corresponding parts on the boundary. We prove the continuity and coercivity of the formulation by employing the technique of operational calculus in the Laplace domain.

math.NA↗

arXiv2019

Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems

We present a stability and convergence theory for the lossy Helmholtz equation and its Galerkin discretization. The boundary conditions are of Robin type. All estimates are explicit with respect to the real and imaginary part of the complex wave number $ζ\in\mathbb{C}$, $\operatorname{Re}ζ\geq0$, $\left\vert ζ\right\vert \geq1$. For the extreme cases $ζ\in\operatorname*{i}\mathbb{R}$ and $ζ\in\mathbb{R}_{\geq0}$, the estimates coincide with the existing estimates in the literature and exhibit a seamless transition between these cases in the right complex half plane.

math.NA↗

arXiv2018

A Fully Discrete Galerkin Method for Abel-type Integral Equations

In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in ractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.

math.NA↗

arXiv2017

A Family of Crouzeix-Raviart Finite Elements in 3D

In this paper we will develop a family of non-conforming "Crouzeix-Raviart" type finite elements in three dimensions. They consist of local polynomials of maximal degree $p\in\mathbb{N}$ on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space.

math.NA↗

arXiv2013

The intersection of bivariate orthogonal polynomials on triangle patches

In this paper, the intersection of bivariate orthogonal polynomials on triangle patches will be investigated. The result is interesting by its own but also has important applications in the theory of a posteriori error estimation for finite element discretizations with $p$-refinement, i.e., if the local polynomial degree of the test and trial functions is increased to improve the accuracy. A triangle patch is a set of disjoint open triangles whose closed union covers a neighborhood of the common triangle vertex. On each triangle we consider the space of orthogonal polynomials of degree n with respect to the weight function which is the product of the barycentric coordinates. We show that the intersection of these polynomial spaces is the null space. The analysis requires the derivation of subtle representations of orthogonal polynomials on triangles. Up to four triangles have to be considered to identify that the intersection is trivial.

math.NA↗

arXiv2013

Functional Estimates for Derivatives of the Modified Bessel Function $K_{0}$ and related Exponential Functions

Let $K_{0}$ denote the modified Bessel function of second kind and zeroth order. In this paper we will studying the function $\tildeω_{n}\left( x\right) :=\frac{\left( -x\right) ^{n}K_{0}^{\left( n\right) }\left( x\right) }{n!}$ for positive argument. The function $\tildeω_{n}$ plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives $\tildeω_{n}^{\left( m\right) }$ with respect to $n$ can be bounded by $O\left( \left( n+1\right) ^{m/2}\right) $ while for small and large arguments $x$ the growth even becomes independent of $n$. These estimates are based on an integral representation of $K_{0}$ which involves the function $g_{n}\left( t\right) =\frac{t^{n}}{n!}\exp\left( -t\right) $ and their derivatives. The estimates then rely on a subtle analysis of $g_{n}$ and its derivatives which we will also present in this paper.

math.CA↗