Abstract

We consider the time-harmonic scalar wave equation in junctions of several different periodic half-waveguides. In general this problem is not well-posed. Several papers propose radiation conditions, i.e., the prescription of the behavior of the solution at the infinities. This ensures uniqueness — except for a countable set of frequencies which correspond to the resonances — and yields existence when one is able to apply Fredholm’s alternative. This solution is called the outgoing solution. However, such radiation conditions are difficult to handle numerically. In this paper, we propose so-called transparent boundary conditions which enable us to characterize the outgoing solution. Moreover, the problem set in a bounded domain containing the junction with these transparent boundary conditions is of Fredholm type. These transparent boundary conditions are based on Dirichlet-to-Neumann operators whose construction is described in the paper. Contrary to the other approaches, the advantage of this approach is that a numerical method can be naturally derived in order to compute the outgoing solution. Numerical results illustrate and validate the method.

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