J. M. Melenk

activity

20132026

most citedRobust exponential convergence of

h p

-FEM in balanced norms for singularly perturbed reaction-diffusion equations

40 citations · 75 across the 18 of their papers we have counts for

collaborators

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13 papers · 1 filter

math.NA2026

Wavenumber-explicit $h p$ -FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media

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 mul…

math.NA2026

Exponential Convergence of $h p$ -FEM for the Integral Fractional Laplacian on cuboids

Björn Bahr, Markus Faustmann, Carlo Marcati +2

For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product $h p$ -finite element approximations on $(0, 1)^{3}$ , for forcing $f$ that is an…

math.NA2024

Parsimonious convolution quadrature

Jens M. Melenk, Jörg Nick

We present a method to rapidly approximate convolution quadrature (CQ) approximations, based on a piecewise polynomial interpolation of the Laplace domain operator, which we call t…

math.NA2024

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

C. Lin, J. M. Melenk, S. Sauter

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…

math.NA2024

Optimal convergence rates in $L^{2}$ for a first order system least squares finite element method -- Part II: inhomogeneous Robin boundary conditions

Maximilian Bernkopf, Jens Markus Melenk

We consider divergence-based high order discretizations of an $L^{2}$ -based first order system least squares formulation of a second order elliptic equation with Robin boundary condi…

math.NA2024

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

C. Lin, J. M. Melenk, S. Sauter

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 t…

math.NA2024

An implementation of hp-FEM for the fractional Laplacian

Björn Bahr, Markus Faustmann, Jens Markus Melenk

We consider the discretization of the $1 d$ -integral Dirichlet fractional Laplacian by $h p$ -finite elements. We present quadrature schemes to set up the stiffness matrix and load ve…

math.NA2024

FEM-BEM coupling for the high-frequency Helmholtz problem

Jens Markus Melenk, Ilaria Perugia, Alexander Rieder

We present a wavenumber-explicit analysis of FEM-BEM coupling methods for time-harmonic Helmholtz problems proposed in arXiv:2004.03523 for conforming discretizations and in arXiv:…

math.NA2024

Optimally truncated WKB approximation for the 1D stationary Schrödinger equation in the highly oscillatory regime

Anton Arnold, Christian Klein, Jannis Körner +1

This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the highly oscillatory regime. We compute an approximate so…

math.NA2023★ 1 cited

Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schrödinger equation

Jannis Körner, Anton Arnold, Christian Klein +1

We discuss the numerical solution of initial value problems for $ε^{2} φ^{''} + a (x) φ = 0$ in the highly oscillatory regime, i.e., with $a (x) > 0$ and $0 < ε ≪ 1$ . We an…

math.NA2014★ 40 cited

Robust exponential convergence of $h p$ -FEM in balanced norms for singularly perturbed reaction-diffusion equations

Jens Markus Melenk, Christos Xenophontos

The $h p$ -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…

math.NA2014★ 8 cited

Simultaneous quasi-optimal convergence in FEM-BEM coupling

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…

math.NA2013★ 15 cited

Existence of $H$ -matrix approximants to the inverses of BEM matrices: the simple-layer operator

Markus Faustmann, Jens Markus Melenk, Dirk Praetorius

We consider the question of approximating the inverse $W = V^{- 1}$ of the Galerkin stiffness matrix $V$ obtained by discretizing the simple-layer operator $…

Wavenumber-explicit hp-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media

Exponential Convergence of hp-FEM for the Integral Fractional Laplacian on cuboids

Parsimonious convolution quadrature

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

Optimal convergence rates in L2 for a first order system least squares finite element method -- Part II: inhomogeneous Robin boundary conditions

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

An implementation of hp-FEM for the fractional Laplacian

FEM-BEM coupling for the high-frequency Helmholtz problem

Optimally truncated WKB approximation for the 1D stationary Schrödinger equation in the highly oscillatory regime

Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schrödinger equation

Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion equations

Simultaneous quasi-optimal convergence in FEM-BEM coupling

Existence of H-matrix approximants to the inverses of BEM matrices: the simple-layer operator

Wavenumber-explicit $h p$ -FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media

Exponential Convergence of $h p$ -FEM for the Integral Fractional Laplacian on cuboids

Optimal convergence rates in $L^{2}$ for a first order system least squares finite element method -- Part II: inhomogeneous Robin boundary conditions

Robust exponential convergence of $h p$ -FEM in balanced norms for singularly perturbed reaction-diffusion equations

Existence of $H$ -matrix approximants to the inverses of BEM matrices: the simple-layer operator