An Exact Bounded Perfectly Matched Layer for Time-Harmonic Scattering Problems

Abstract

Abstract. The aim of this paper is to introduce an "exact" bounded perfectly matched layer (PML) for the scalar Helmholtz equation. This PML is based on using a non integrable absorbing function. "Exactness" must be understood in the sense that this technique allows exact recovering of the solution to time-harmonic scattering problems in unbounded domains. In spite of the singularity of the absorbing function, the coupled fluid/PML problem is well posed when the solution is sought in an adequate weighted Sobolev space. The resulting variational formulation can be numerically dealt with standard finite elements. The high accuracy of this approach is numerically demonstrated as compared with a classical PML technique. 1. Introduction. The typical first step for the numerical solution by finite elements or finite differences of any scattering problem in an unbounded domain is to truncate the computational domain, which entails an inherent difficulty: to choose boundary conditions to replace the Sommerfeld radiation condition at infinity (see for instance [21]).There are several techniques to deal with this: boundary element methods, infinite element methods, Dirichlet-to-Neumann methods based on Fourier expansions, or the use of absorbing boundary conditions. The potential advantages of each of them have been widely studied in the literature [25,35,18].If the domain of the original problem is truncated with a sphere, then the Dirichletto-Neumann (DtN) boundary condition is exactly known (see [21,32]). However, this boundary condition involves an infinite series which must be truncated for numerical use, thus introducing errors. Moreover, the exact DtN condition is non local, leading to dense blocks in the linear system to be solved, when a finite element method is used.As a partial solution to these drawbacks, local absorbing boundary conditions (ABCs) can be introduced to preserve the computational efficiency of the numerical method. Those of Bayliss and Turkel [5], Engquist and Majda [19], and Feng [20] are among the most widely used. However, in spite of the simple implementation of lowest order ABCs, good accuracy is only achieved for higher order ones [37], because these conditions are not fully non-reflecting on the truncated boundary of

Two scale Hardy space infinite elements for scalar waveguide problems

“…Since all resonances on the real axis are independent of s 0 , the identity theorem for holomorphic functions yields: If ω is a resonance for one s (1) 0 , then it is also a resonance for a second s (2) 0 if there exists a path in the complex plane connecting the real axis with ω such that κ s (1) 0 n and κ s (2) 0 n are holomorphic in a neighborhood of this path. In other words, for all ω on this path iκ s (1) 0 n and iκ s (2) 0 n have to stay in + and are therefore not allowed to intersect .…”

“…Remark 3.1 In [2] an exact complex scaling is proposed, where a truncation of the infinite layer is not needed. This can be achieved by replacing the linear complex scaling (2.8) with a more general scaling of the formξ = σ (ξ) such that lim ξ →L (ξ) = ∞ for a finite L > 0.…”

“…Using the scaling proposed in [2] with (ξ )/ (ξ) → 0 for ξ → L enforces wavenumbers κ n with (κ n ) ≥ 0. Hence the wavenumbers, which are implicitly chosen by such a complex scaling, are no longer holomorphic in a neighborhood of the positive real axis, since (κ n (ω)) jumps for all n with √ λ n > ω if ω > 0 is extended to complex ω with (ω) < 0.…”

Perfectly Matched Layers for Time-Harmonic Second Order Elliptic Problems

“…Furthermore, in spite of the fact that the PML has been originally settled in Cartesian coordinates by Berenger, Collino and Monk [28] proposed a similar complex-valued change of coordinates to build a PML in curvilinear coordinates, as we have also emphasized in [19].…”

Exact nonreflecting boundary conditions for three dimensional poroelastic wave equations

“…It has been widely used for electromagnetic and elastic wave equations (e.g. [11,18,41,48]), poroelastic elastic equations [58,59], and the mixed hyperbolic-parabolic systems [3]. The comparison of high-order absorbing boundary conditions and PML can be found in [52].…”