Theoretical considerations on the computation of generalized timeperiodic waves

We present both theory and an algorithm for solving time-harmonic wave prob-lems in a general setting. The time-harmonic solutions will be achieved by com-puting time-periodic solutions of the original wave equations. Thus, an exact con-trollability technique is proposed to solve the time-dependent wave equations. We discuss a first order Maxwell type system, which will be formulated in the frame-work of alternating differential forms. This enables us to investigate different kinds of classical wave problems in one fell swoop, such as acoustic, electro-magnetic or elastic wave problems. After a sufficient theory is established, we formulate our ex-act controllability problem and suggest a least-squares optimization procedure for its solution, which itself is solved in a natural way by a conjugate gradient algo-rithm operating in a purely L2-type Hilbert space. Therefore, it might be one of the biggest advances of this approach that the proposed conjugate gradient algorithm does not need any preconditioning