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Stefan Sauter

Publications and source records attributed to Stefan Sauter.

16 recordsLinked to original sources
arXiv2026

Computation of whispering gallery modes for spherical symmetric, heterogeneous Helmholtz problems with piecewise smooth refractive index

In this paper, we develop a numerical method for the computation of (quasi-)resonances in spherical symmetric, heterogeneous Helmholtz problems with piecewise smooth refractive index. Our focus lies in resonances very close to the real axis, which characterize the so-called whispering gallery modes. Our method involves a modal equation incorporating fundamental solutions to decoupled problems, extending the known modal equation to the case of piecewise smooth coefficients. We first establish the well-posedness of the fundamental system, then we formulate the problem of resonances as a nonlinear eigenvalue problem, whose determinant will be the modal equation in the piecewise smooth case. In combination with the numerical approximation of the fundamental solutions using a spectral method, we propose a Newton method to solve the nonlinear modal equation with a proper scaling. We prove the local convergence of the algorithm in the piecewise constant case by showing the simplicity of the roots. We confirm our approach through a series of numerical experiments in the piecewise constant as well as in the variable case.

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arXiv2025

Stable skeleton integral equations for general coefficient Helmholtz transmission problems

A novel variational formulation of layer potentials and boundary integral operators generalizes their classical construction by Green's functions, which are not explicitly available for Helmholtz problems with variable coefficients. Wavenumber explicit estimates and properties like jump conditions follow directly from their variational definition and enable a non-local (``integral'') formulation of acoustic transmission problems (TP) with piecewise Lipschitz coefficients. We obtain the well-posedness of the integral equations directly from the stability of the underlying TP. The simultaneous analysis for general dimensions and complex wavenumbers (in this paper) imposes an artificial boundary on the external Helmholtz problem and employs recent insights into the associated Dirichlet-to-Neumann map.

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arXiv2025

A spectral boundary element method for acoustic interference problems

In this paper we consider high-frequency acoustic transmission problems with jumping coefficients modelled by Helmholtz equations. The solution then is highly oscillatory and, in addition, may be localized in a very small vicinity of interfaces (whispering gallery modes). For the reliable numerical approximation a) the PDE is tranformed in a classical single trace integral equation on the interfaces and b) a spectral Galerkin boundary element method is employed for its solution. We show that the resulting integral equation is well posed and analyze the convergence of the boundary element method for the particular case of concentric circular interfaces. We prove a condition on the number of degrees of freedom for quasi-optimal convergence. Numerical experiments confirm the efficiency of our method and the sharpness of the theoretical estimates.

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arXiv2024

Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions

In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $\left( d-1\right) $ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and $k>1$.

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arXiv2022

On the Inf-Sup Stability of Crouzeix-Raviart Stokes Elements in 3D

We consider non-conforming discretizations of the stationary Stokes equation in three spatial dimensions by Crouzeix-Raviart type elements. The original definition in the seminal paper by M. Crouzeix and P.-A. Raviart in 1973 is implicit and also contains substantial freedom for a concrete choice. In this paper, we introduce basic Crouzeix-Raviart basis functions in 3D in analogy to the 2D case in a fully explicit way. We prove that this basic CrouzeixRaviart element for the Stokes equation is inf-sup stable for polynomial degree $k =2$ (quadratic velocity approximation). We identify spurious pressure modes for the conforming $(k; k - 1)$ 3D Stokes element and show that these are eliminated by using the basic Crouzeix-Raviart space.

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arXiv2022

Stabilized leapfrog based local time-stepping method for the wave equation

Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. In \cite{DiazGrote09}, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step, $Δt$, depends on the smallest elements in the mesh \cite{grote_sauter_1}. In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain discrete values of $Δt$. To remove those critical values of $Δt$, we apply a slight modification (as in recent work on LF-Chebyshev methods \cite{CarHocStu19}) to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate, satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition where $Δt$ no longer depends on the mesh size inside the locally refined region.

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arXiv2022

Solvability of Discrete Helmholtz Equations

We study the unique solvability of the discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin $hp$-finite element method. Well-posedness of the discrete equations is typically investigated by applying a compact perturbation to the continuous Helmholtz problem so that a "sufficiently rich" discretization results in a "sufficiently small" perturbation of the continuous problem and well-posedness is inherited via Fredholm's alternative. The qualitative notion "sufficiently rich", however, involves unknown constants and is only of asymptotic nature. Our paper is focussed on a fully discrete approach by mimicking the tools for proving well-posedness of the continuous problem directly on the discrete level. In this way, a computable criterion is derived which certifies discrete well-posedness without relying on an asymptotic perturbation argument. By using this novel approach we obtain a) new stability results for the $hp$-FEM for the Helmholtz problem b) examples for meshes such that the discretization becomes unstable (stiffness matrix is singular), and c) a simple checking Algorithm MOTZ "marching-of-the-zeros" which guarantees in an a posteriori way that a given mesh is certified for a stable Helmholtz discretization.

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arXiv2020

The Heterogeneous Helmholtz Problem with Spherical Symmetry: Green's Operator and Stability Estimates

We study wave propagation phenomena modelled in the frequency domain by the Helmholtz equation in heterogeneous media with focus on media with discontinuous, highly oscillating wave speed. We restrict to problems with spherical symmetry and will derive explicit representations of the Green's operator and stability estimates which are explicit in the frequency and the wave speed.

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arXiv2019

Numerical approximation of Poisson problems in long domains

In this paper, we consider the Poisson equation on a "long" domain which is the Cartesian product of a one-dimensional long interval with a (d-1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by an alternating-least-squares (ALS) type method (method 2). Numerical experiments show that the ALS is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretised differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

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arXiv2019

Benchmark computation of eigenvalues with large defect for non-selfadjoint elliptic differential operators

In this paper we present benchmark problems for non-selfadjoint elliptic eigenvalue problems with large defect and ascent. We describe the derivation of the benchmark problem with a discontinuous coefficient and mixed boundary conditions. Numerical experiments are performed to investigate the convergence of a Galerkin finite element method with respect to the discretization parameters, the regularity of the problem, and the ascent of the eigenvalue. This allows us to verify the sharpness of the theoretical estimates from the literature with respect to these parameters. We provide numerical evidence about the size of the ascent and show that it is important to consider the mean value for the eigenvalue approximation.

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arXiv2019

Variable Order, Directional H2-Matrices for Helmholtz Problems with Complex Frequency

The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence their sparse approximation is of outstanding importance. In our paper we will generalize the directional $\mathcal{H}^{2}$-matrix techniques from the \textquotedblleft pure\textquotedblright\ Helmholtz operator $\mathcal{L}u=-Δu+ζ^{2}u$ with $ζ=-\operatorname*{i}k$, $k\in\mathbb{R}$, to general complex frequencies $ζ\in\mathbb{C}$ with $\operatorname{Re}ζ>0$. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition which contains $\operatorname{Re}ζ$ in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent \textit{directional expansion functions}. We develop an error analysis which is explicit with respect to the expansion order and with respect to $\operatorname{Re}ζ$ and $\operatorname{Im}ζ$. This allows to choose the \textit{variable }expansion order in a quasi-optimal way depending on $\operatorname{Re}ζ$ but independent of, possibly large, $\operatorname{Im}ζ$. The complexity analysis is explicit with respect to $\operatorname{Re}ζ$ and $\operatorname{Im}ζ$ and shows how higher values of $\operatorname{Re}% ζ$ reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.

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arXiv2018

Stability estimate for the Helmholtz equation with rapidly jumping coefficients

The goal of this paper is to investigate the stability of the Helmholtz equation in the high- frequency regime with non-smooth and rapidly oscillating coefficients on bounded domains. Existence and uniqueness of the problem can be proved using the unique continuation principle in Fredholm's alternative. However, this approach does not give directly a coefficient-explicit energy estimate. We present a new theoretical approach for the one-dimensional problem and find that for a new class of coefficients, including coefficients with an arbitrary number of discontinuities, the stability constant (i.e., the norm of the solution operator) is bounded by a term independent of the number of jumps. We emphasize that no periodicity of the coefficients is required. By selecting the wave speed function in a certain \resonant" way, we construct a class of oscillatory configurations, such that the stability constant grows exponentially in the frequency. This shows that our estimates are sharp.

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arXiv2017

Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In [18] a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here a rigorous convergence proof is presented for the fully-discrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.

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arXiv2015

A Posteriori Error Estimation of hp-dG Finite Element Methods for Highly Indefinite Helmholtz Problems (extended version)

In this paper, we will consider an $hp$-finite elements discretization of a highly indefinite Helmholtz problem by some dG formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters $h$ and $p$. In contrast to the conventional conforming finite element method for indefinite problems, the dG formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh. Numerical experiments will illustrate the efficiency and robustness of the method.

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arXiv2014

Adaptive Time Discretization for Retarded Potentials

In this paper, we will present advanced discretization methods for solving retarded potential integral equations. We employ a $C^{\infty}$-partition of unity method in time and a conventional boundary element method for the spatial discretization. One essential point for the algorithmic realization is the development of an efficient method for approximation the elements of the arising system matrix. We present here an approach which is based on quadrature for (non-analytic) $C^{\infty}$ functions in combination with certain Chebyshev expansions. Furthermore we introduce an a posteriori error estimator for the time discretization which is employed also as an error indicator for adaptive refinement. Numerical experiments show the fast convergence of the proposed quadrature method and the efficiency of the adaptive solution process.

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