The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator

Abstract

When a problem is posed on an unbounded domain, the domain needs to be truncated in order to perform computations, and the pole condition is a new technique developed over the last few years for this purpose. The subject of domain truncation is already an established research field. It was started in 1977 in a seminal paper by Enquist and Majda [6], where a systematic method to obtain absorbing boundary conditions (ABCs) is introduced for wave propagation phenomena. Absorbing boundary conditions are approximations of transparent boundary conditions (TBCs), which, when used to truncate the unbounded domain, lead by definition precisely to the restriction of the original solution on the unbounded domain. Unfortunately transparent boundary conditions often involve expensive non-local operators and are thus inconvenient. Absorbing boundary conditions became immediately a field of interest of mathematicians in approximation theory, see for example [3, 9]. Recent reviews on non-reflecting or absorbing boundary conditions concerning the wave equation are [8] by Hagstrom and more recently Givoli [7]. Non-reflecting boundary conditions for the transient Schrodinger equation are reviewed by Antoine et al. [1].