The proposed reserch develops and analyzes efficient numerical methods for time-harmonic acoustic wave propagation problems in fluids and solids. Particularly, exterior two-dimensional and three-dimensional problems for acoustic scattering in inhomogenous media and fluid-structure scattering are considered. The discretization of the governing PDE models is performed using finite elements on orthogonal rectangular meshes with local adaptation to boundaries and interfaces. Special phase error reducing low-order finite elements are studied and employed. The aim is to solve three-dimensional time-harmonic wave propagation problems in which the diameter of the computational domain is from tens to hundreds wavelengths. These problems lead to very large systems of linear equations which can have from millions to several billions unknowns. Numerical methods for this frequency range are under active research, and no efficient method still seem to exist. The large scale linear systems are solved iteratively with a Schwarz-type domain decomposition preconditioner. Very efficient subdomain preconditioners are constructed using fast direct solvers by means of domain embedding. This novelty makes the proposed approach very fast. The conditioning of preconditioned systems are analyzed for suitable model problems. The resulting methods are constructed in such a way that the iterations can be carried out on a small sparse subspace related to the interfaces.
The studied scattering problems arise routinely in many disciplines. Due to limitations of contemporary methods and computational resources many of these problems cannot be solved in practice. The aim of this research is to develop numerical methods which enable the solution of many of these large problems. The first class of studied model problems are acoustical geological surveys employed in oil exploration. Currently it is necessary to use crude methods for surveys which are in many cases inaccurate. The developed methods can eventually lead to improved yield in the extraction of oil and reduced environmental impact. The second class of model problems describe scattering by elastic objects like mines in seabed. Mines have been the single largest cause of fatalities in naval warfare. This part of the research is conducted in cooperation with researchers at a Navy research center and it is closely related to a project funded by the Office of Naval Research. A broader goal of this activity is develop computational methods which help to enhance the detection and identification capabilities of mines in seabed.
