Mathematics – Analysis of PDEs
Scientific paper
2011-05-20
Mathematics
Analysis of PDEs
Key Words: wave equation, Maxwell's equations, differential forms, differential geometry, time-periodic waves, time-harmonic w
Scientific paper
We present both, theory and an algorithm for solving time-harmonic wave problems in a general setting. The time-harmonic solutions will be achieved by computing time-periodic solutions of the original wave equations. Thus, an exact controllability technique is proposed to solve the time-dependent wave equations. We discuss a first order Maxwell type system, which will be formulated in the framework 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 exact controllability problem and suggest a least-squares optimization procedure for its solution, which itself is solved in a natural way by a conjugate gradient algorithm operating in the canonical 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.
Pauly Dirk
Rossi Tuomo
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