A Galerkin Boundary Element Method for a High Frequency Scattering Problem

Abstract

In this paper we consider the numerical solution of the Helmholtz equation

$$ \Delta u + {k^2}u = 0 $$

((1))

in the upper half-plane U:= {(x _i, x ₂) ∈ R ²: x ₂ > 0}, with impedance boundary condition

$$ \frac{{\partial u}}{{\partial {x_2}}} + ik\beta u = f $$

((2))

on Г:= {(x ₁,0): x ₁ ∈ R}, where k > 0 (the wavenumber) is some arbitrary positive constant. This boundary value problem can arise when modelling the acoustic scattering of an incident wave by a planar surface with spatially varying acoustical properties [1]. The total acoustic field u ^t ∈ C(Ū) ∩C ²(U) satisfies (l)–(2) where the wavenumber k = 2πμ/c, with μ being the frequency of the incident wave, c the speed of sound in U, and f ≡ 0. For simplicity of exposition, here we restrict our attention to the case of plane wave incidence, so that the incident field uⁱ is given by

$$ {u^i}(x) = {e^{{ikx \cdot d}}},\quad d({d_1},{d_2}) = (\sin \theta, - \cos \theta ) $$

((3))

with θ ∈ (—π/2, π/2) being the angle of incidence. The reflected or scattered part of the wave field is u ∈ C(Ū) ∩ C ²(U), defined by u = u ^t - u ⁱ. The scattered field then also satisfies (1)–(2) with

$$ f(s): = ( - \partial {u^i}/\partial {x_2} - ik\beta {u^i})(s,0) = ik{e^{{iks\sin \theta }}}(\cos \theta - \beta (s)),\quad s \in R $$

((4))