Complex riemannian metric and absorbing boundary conditions

In computational electromagnetic and acoustic scattering, the unbounded Euc lidean space R-3 is often modeled by a bounded domain with an absorbing bou ndary condition. One possible approach to create such absorbing boundary co ndition is to surround the computational domain by a non-reflecting artific ial sponge layer that absorbs quickly the scattered waves. This approach is called the method of a Perfectly Matched Layer (PML). In this paper we pro ve that such absorbing boundary layers can be obtained by using complex Rie mannian metric g(ij). We show that the boundary layer is non-reflecting whe n g is flat, that is, the curvature tensor of the complex metric gij is zer o. This fact gives an invariant formulation for the absorbing boundary laye rs as well as give us new kind of absorbing boundary layers for Maxwell and Helmholtz equations. Moreover, we show that all Perfectly Matched Layers, that is, absorbing boundary layers obtained through a complexification of c oordinates corresponds to flat complex manifolds. Finally, we discuss the r elation of the absorbing boundary layers and the complex scaling technique, developed by Sjostrand and Zworski for the study of scattering poles. (C) 2001 Editions scientifiques et medicales Elsevier SAS.