Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves

by   Jeffrey Galkowski, et al.

We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e. away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.



There are no comments yet.


page 9

page 10

page 11

page 13


A sharp relative-error bound for the Helmholtz h-FEM at high frequency

For the h-finite-element method (h-FEM) applied to the Helmholtz equatio...

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

We study a commonly-used second-kind boundary-integral equation for solv...

Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

We consider GMRES applied to discretisations of the high-frequency Helmh...

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

We prove quantitative norm bounds for a family of operators involving im...

A simple framework for arriving at bounds on effective moduli in heterogeneous anisotropic poroelastic solids

The concepts of representative volume element (RVE), statistical homogen...

Highly accurate acoustic scattering: Isogeometric Analysis coupled with local high order Farfield Expansion ABC

This work is concerned with a unique combination of high order local abs...

Domain decomposition preconditioners for high-order discretisations of the heterogeneous Helmholtz equation

We consider one-level additive Schwarz domain decomposition precondition...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.