Narrow Results

We consider a controllability method for the time-periodic solution of the two-dimensional scalar wave equation with a first order absorbing boundary condition describing the scattering of a time-harmonic incident wave by a sound-soft obstacle. Solution of the time-harmonic equation is equivalent to finding a periodic solution for the corresponding time-dependent wave equation. We formulate the problem as an exact controllability one and solve the wave equation in time-domain. In a mixed formulation we look for solutions u = (v, p)^T. The use of mixed formulation allows us to set the related controllability problem in (L²(Ω))^d+1, a space of square-integrable functions in dimension d + 1. No preconditioning is needed when solving this with conjugate gradient method. We present numerical results concerning performance and convergence properties of the method.