Jens Markus Melenk

Sofi Esterhazy, Jens Markus Melenk

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.

Markus Aurada, Michael Feischl, Thomas Führer et al.

We consider a (possibly) nonlinear interface problem in 2D and 3D, which is solved by use of various adaptive FEM-BEM coupling strategies, namely the Johnson-Nédélec coupling, the Bielak-MacCamy coupling, and Costabel's symmetric coupling. We provide a framework to prove that the continuous as well as the discrete Galerkin solutions of these coupling methods additionally solve an appropriate operator equation with a Lipschitz continuous and strongly monotone operator. Therefore, the coupling formulations are well-defined, and the Galerkin solutions are quasi-optimal in the sense of a Céa-type lemma. For the respective Galerkin discretizations with lowest-order polynomials, we provide reliable residual-based error estimators. Together with an estimator reduction property, we prove convergence of the adaptive FEM-BEM coupling methods. A key point for the proof of the estimator reduction are novel inverse-type estimates for the involved boundary integral operators which are advertized. Numerical experiments conclude the work and compare performance and effectivity of the three adaptive coupling procedures in the presence of generic singularities.

Steffen Börm, Jens Markus Melenk

We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and postmultiplication by plane waves. It is shown to converge exponentially in the polynomial degree and supports multilevel approximation techniques. Our convergence analysis may be employed to establish exponential convergence of certain classes of fast methods for discretizations of the Helmholtz integral operator that feature polylogarithmic-linear complexity.

Markus Faustmann, Jens Markus Melenk, Dirk Praetorius

For the discretization of the integral fractional Laplacian $(-Δ)^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.

Jens Markus Melenk, Tobias Wurzer

For the reference triangle or tetrahedron $T$, we study the stability properties of the $L^2(T)$-projection $Π_N$ onto the space of polynomials of degree $N$. We show $\|Π_N u\|_{L^2(\partial T)}^2 \leq C \|u\|_{L^2(T)} \|u\|_{H^1(T)}$. This implies optimal convergence rates for the approximation error $\|u - Π_N u\|_{L^2(\partial T)}$ for all $u \in H^k(T)$, $k > 1/2$.

Jens Markus Melenk, Christos Xenophontos, Lisa Oberbroeckling

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< ε\le μ\le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $ε$ and $% μ$. We construct full asymptotic expansions together with error bounds that cover the complete range $0 < ε\leq μ\leq 1$. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.

Jens Markus Melenk, Alexander Rieder

We propose a numerical scheme to solve the time dependent linear Schrödinger equation. The discretization is carried out by combining a Runge-Kutta time-stepping scheme with a finite element discretization in space. Since the Schrödinger equation is posed on the whole space $\R^d$ we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.

Jens Markus Melenk, Markus Faustmann

The hp-version of the finite element method is applied to singularly perturbed reaction-diffusion type equations on polygonal domains. The solution exhibits boundary layers as well as corner layers. On a class of meshes that are suitable refined near the boundary and the corners, robust exponential convergence (in the polynomial degree) is shown in both a balanced norm and the maximum norm.

Markus Faustmann, Jens Markus Melenk, Dirk Praetorius

We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block rank. We cover in particular the data-sparse format of $\mathcal{H}$-matrices. We show the approximability result for two types of discretizations. The first one is a saddle point formulation, which incorporates the constraint of vanishing mean of the solution. The second discretization is based on a stabilized hyper-singular operator, which leads to symmetric positive definite matrices. In this latter setting, we also show that the hierarchical Cholesky factorization can be approximated at an exponential rate in the block rank.

Markus Aurada, Michael Feischl, Thomas Führer et al.

We prove inverse-type estimates for the four classical boundary integral operators associated with the Laplace operator. These estimates are used to show convergence of an h-adaptive algorithm for the coupling of a finite element method with a boundary element method which is driven by a weighted residual error estimator.

Markus Faustmann, Jens Markus Melenk

The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a priori estimates in $L^2$ for Symm's integral equation and in $H^1$ for the hypersingular equation. The local rate of convergence is limited by the local regularity of the sought solution and the sum of the global regularity and additional regularity provided by the shift theorem for a dual problem.

Maximilian Bernkopf, Jens Markus Melenk

Extending the wavenumber-explicit analysis of [Chen & Qiu, J. Comput. Appl. Math. 309 (2017)], we analyze the $L^2$-convergence of a least squares method for the Helmholtz equation with wavenumber $k$. For domains with an analytic boundary, we obtain improved rates in the mesh size $h$ and the polynomial degree $p$ under the scale resolution condition that $hk/p$ is sufficiently small and $p/\log k$ is sufficiently large.

Jens Markus Melenk, Dirk Praetorius, Barbara Wohlmuth

We consider the symmetric FEM-BEM coupling that connects two linear elliptic second order partial differential equations posed in a bounded domain $Ω$ and its complement, where the exterior problem is restated by an integral equation on the coupling boundary $Γ=\partialΩ$. We assume that the corresponding transmission problem admits a shift theorem for data in $H^{-1+s}$, $s \in [-1,-1+s_0]$, $s_0 > 1/2$. We analyze the discretization by piecewise polynomials of degree $k$ for the domain variable and piecewise polynomials of degree $k-1$ for the flux variable on the coupling boundary. Given sufficient regularity we show that (up to logarithmic factors) the optimal convergence $O(h^{k+1/2})$ in the $H^{-1/2}(Γ)$-norm is obtained for the flux variable, while classical arguments by Céa-type quasi-optimality and standard approximation results provide only $O(h^k)$ for the overall error in the natural product norm on $H^1(Ω)\times H^{-1/2}(Γ)$.

Jens Markus Melenk, David Wörgötter

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 $k$.

Jens Markus Melenk, David Wörgötter

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.

Björn Bahr, Markus Faustmann, Carlo Marcati et al.

For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product $hp$-finite element approximations on $(0,1)^3$, for forcing $f$ that is analytic in $[0,1]^3$. Exploiting analytic regularity estimates in weighted Sobolev spaces, we prove for $hp$-GLL interpolation approximations with $N$ degrees of freedom the energy norm error bound $\lesssim \exp(-b\sqrt[6]{N})$. Tensor product mesh families which are geometrically refined towards all sides of $(0,1)^3$ are used. Numerical experiments with $hp$-Galerkin FEM confirm the bound.

Michael Karkulik, Jens Markus Melenk

The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasi-uniform mesh. We show that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.

Steffen Börm, Christina Börst, Jens Markus Melenk

Butterfly algorithms are an effective multilevel technique to compress discretizations of integral operators with highly oscillatory kernel functions. The particular version of the butterfly algorithm considered here realizes the transfer between levels by Chebyshev interpolation. We present a refinement of the analysis that improves the stability estimates underlying the error bounds.

Michael Karkulik, Jens Markus Melenk

We develop a regularization operator based on smoothing on a locally defined length scale. This operator is defined on $L_1$ and has approximation properties that are given by the local regularity of the function it is applied to and the local length scale. Additionally, the regularized function satisfies inverse estimates commensurate with the approximation orders. By combining this operator with a classical hp-interpolation operator, we obtain an hp-Clément type quasi-interpolation operator, i.e., an operator that requires minimal smoothness of the function to be approximated but has the expected approximation properties in terms of the local mesh size and polynomial degree. As a second application, we consider residual error estimates in hp-boundary element methods that are explicit in the local mesh size and the local approximation order.

Thomas Führer, Jens Markus Melenk, Dirk Praetorius et al.

We propose and analyze an overlapping Schwarz preconditioner for the $p$ and $hp$ boundary element method for the hypersingular integral equation in 3D. We consider surface triangulations consisting of triangles. The condition number is bounded uniformly in the mesh size $h$ and the polynomial order $p$. The preconditioner handles adaptively refined meshes and is based on a local multilevel preconditioner for the lowest order space. Numerical experiments on different geometries illustrate its robustness.

Markus Aurada, Michael Feischl, Thomas Führer et al.

We prove local inverse-type estimates for the four non-local boundary integral operators associated with the Laplace operator on a bounded d-dimensional Lipschitz domain Omega for d >= 2 with piecewise smooth boundary. For piecewise polynomial ansatz spaces and d = 2 or 3, the inverse estimates are explicit in both the local mesh width and the approximation order. An application to efficiency estimates in a posteriori error estimation in boundary element methods is given.

Jens Markus Melenk, Christos Xenophontos

The $hp$-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed in one- and two-dimensional domains with analytic boundary. On suitably designed \emph{Spectral Boundary Layer meshes}, robust exponential convergence in a balanced norm is shown. This balanced norm is stronger than the energy norm in that the boundary layers are $O(1)$ uniformly in the singular perturbation parameter. Robust exponential convergence in the maximum norm is also established. The theoretical findings are illustrated with two numerical experiments.

Markus Faustmann, Jens Markus Melenk, Dirk Praetorius

We consider the question of approximating the inverse $\mathbf W = \mathbf V^{-1}$ of the Galerkin stiffness matrix $\mathbf V$ obtained by discretizing the simple-layer operator $V$ with piecewise constant functions. The block partitioning of $\mathbf W$ is assumed to satisfy any of the standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator $\mathbf V$. We show that $\mathbf W$ can be approximated by blockwise low-rank matrices such that the error decays exponentially in the block rank employed. Similar exponential approximability results are shown for the Cholesky factorization of $\mathbf V$.