Complex Riemannian metric and absorbing boundary conditions
The scattering of electromagnetic or acoustic waves from a given obstacle takes place in an unbounded space which causes difficulties in numerical simulations. One possible approach is to surround the scatterer by an absorbing boundary layer. The common PLM (perfectly matched layer) method consists in embedding of the original space \(\mathbb{R}^3\) into \(\mathbb{C}^3\), the complex stretching of coordinates. The authors propose a different and more geometrical approach by using complex Riemannian metric. This provides a new kind of absorbing boundary layers for Maxwell and Helmholtz equations. A pseudo-Riemannian manifold \((M,g)\) is called an absorbing manifold if \(g\) is flat and nondegenerate complex metric such that there is an isometry \(M\to\mathbb{C}^3\) on a totally real submanifold of \(\mathbb{C}^3\) but \(g\) is the common Euclidean metric on a relatively compact open subset \(D\subset M\). If we place a source or scatterer inside \(D\), the waves within \(D\) coincide with the scattering solution while outside \(D\), the complex metric causes reflectionless attenuation of exponential type. The authors carefully introduce all geometrical prerequisites and thoroughly deal with the existence proofs of solutions on absorbing manifolds.
scientific article; zbMATH DE number 1770845