Related papers: Sharp high-frequency estimates for the Helmholtz e…
In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of…
A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated…
The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation…
Consider three-dimensional fractional MHD equations in an exterior domain with the Dirichlet boundary condition assumed. Asymptotic behaviours of weak solutions to the three-dimensional exterior fractional MHD equations are studied. $L^2$…
We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star-shaped domain in $\mathbb{R}^d$, $d\ge 2$. Conventional boundary-based methods require a…
We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under…
We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We…
This paper considers to the problems of diffraction of electromagnetic waves on a half-plane, which has a finite inclusion in the form of a Lipschitz curve. The diffraction problem formulated as boundary value problem for Helmholtz…
We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, $u(\x) - \alpha^2 \Delta u(\x) = 0$, in bounded or unbounded multiply-connected domains. We consider both Dirichlet…
In this paper we study the dependence of the H\"older estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the H\"older regularity of the solutions to the Dirichlet and Neumann problems in the disk…
In the present article we present a particular combination of boundary problems for the inhomogeneous tri-analytic equation: the Neumann-(Dirichlet-Neuman) problem and the (Dirichlet-Neumann)-Dirichlet problem. In order to obtain the…
In this paper, new boundary differential equations for the two-dimensional exterior scattering problem have been derived. It has been shown that the Helmholtz equation can be reduced to an inhomogeneous Bessel's equation in a body-fitted…
This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions for which Dirichlet and Neumann conditions are specified on…
We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and we solve the Neumann problem for the Helmholtz equation both in $\Omega$ and in the exterior of $\Omega$. We look for…
This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This…
It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method,…
We prove decay estimates in the interior for solutions to elliptic equations in divergence form with Lipschitz continuous coefficients. The estimates explicitly depend on the distance from the boundary and on suitable notions of frequency…
This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for…
A Fredholm integral equation of the second kind with the generalized Neumann kernel associated with the Riemann-Hilbert problem on unbounded multiply connected regions will be derived and studied in this paper. The derived integral equation…
We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the…