arXiv subjects

J. M. Melenk

Publications and source records attributed to J. M. Melenk.

6 recordsLinked to original sources

arXiv2025

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

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 the new representation are: a) in addition to additive terms that appear also in the case of Dirichlet or Neumann boundary conditions, the remaining part of the Green's function is factored into an oscillatory complex exponential function (with the product of the wavenumber and the eikonal as argument) and a remaining function which is slowly varying and hence allows for efficient polynomial approximation; b) the representation is given uniformly for all parameters by a single formula which consists of the product of two analytic functions.

math.NA↗

arXiv2024

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

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 an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.

math.NA↗

arXiv2024

Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media

We present a wavenumber-explicit convergence analysis of the hp finite element method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber $k$. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.

math.NA↗

arXiv2017

On thin plate spline interpolation

We present a simple, PDE-based proof of the result [M. Johnson, 2001] that the error estimates of [J. Duchon, 1978] for thin plate spline interpolation can be improved by $h^{1/2}$. We illustrate that ${\mathcal H}$-matrix techniques can successfully be employed to solve very large thin plate spline interpolation problems

math.NA↗

arXiv2015

On optimal $L^2$- and surface flux convergence in FEM (extended version)

We show that optimal $L^2$-convergence in the finite element method on quasi-uniform meshes can be achieved if, for some $s_0 > 1/2$, the boundary value problem has the mapping property $H^{-1+s} \rightarrow H^{1+s}$ for $s \in [0,s_0]$. The lack of full elliptic regularity in the dual problem has to be compensated by additional regularity of the exact solution. Furthermore, we analyze for a Dirichlet problem the approximation of the normal derivative on the boundary without convexity assumption on the domain. We show that (up to logarithmic factors) the optimal rate is obtained.

math.NA↗

arXiv2014

Scalable numerical approach for the steady-state ab initio laser theory

We present an efficient and flexible method for solving the non-linear lasing equations of the steady-state ab initio laser theory. Our strategy is to solve the underlying system of partial differential equations directly, without the need of setting up a parametrized basis of constant flux states. We validate this approach in one-dimensional as well as in cylindrical systems, and demonstrate its scalability to full-vector three-dimensional calculations in photonic-crystal slabs. Our method paves the way for efficient and accurate simulations of lasing structures which were previously inaccessible.

physics.optics↗