Stefan Sauter

Stefan Sauter, Celine Torres

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.

Stefan Sauter, Jakob Zech

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.

Marcus J. Grote, Michaela Mehlin, Stefan Sauter

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.

Steffen Börm, Maria Lopez-Fernandez, Stefan Sauter

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.