Integral equation methods for the Helmholtz equation

Boundary integral methods in high frequency scattering

“…The Galerkin discretisation of this integral equation yields a symmetric stiffness matrix and has no spurious frequencies provided ℜZ > 0. The authors argued (partly referring to earlier results [40]) that, due to the oscillatory solution of (1.1), in general the conventional boundary element approximation v h of this equation, using step-size h and polynomial degree p, would satisfy an…”

Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. II. The discrete problem

“…In general, ph does not belong to Vh(p). So, to compare ph and/?, we first adapt Giroire's results [9] on the scalar Helmholtz equation. This and a kind of stability property given by (2.55) lead to an estimate of the errors Observe that this last relation may be equivalently written as…”

“…[9], [12]), we easily prove where C is a constant independent of h,p,X,c,u>, and ej(x, T) is the function (cf.…”

Consistency Estimates for a Double-Layer Potential and Application to the Numerical Analysis of the Boundary-Element Approximation of Acoustic Scattering by a Penetrable Object

“…Without being exhaustive, we quote: Laplace and Helmholtz equations (cf. Nédélec [24], Giroire [12,13]), scattering of an electromagnetic wave by a dielectric cylinder (cf. Chang and Harrington [4]), coupling interior and boundary elements (cf.…”

“…By contrast, for the single-layer potential, the method initiated by Nédélec [25] gives complete answers in several directions (cf., e.g., Le Roux [22], Giroire [12,13], Bendali [2,3], Johnson and Nédélec [18], etc.). This seems to be due to the following feature.…”