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Boundary Conditions in the Schur Complement Preconditioning of Dissipative Acoustic Equations

This paper is concerned with the development of robust and efficient iterative techniques for solving the coupled systems of linear equations which arise from a mixed finite element discretization of dissipative acoustic equations. The system of linearized, time-harmonic acoustic equations we consider is based on the Navier-Stokes, continuity, and energy equations accompanied by suitable equations of state. In this paper, we focus on the design of preconditioners that can be used in combination with the Krylov subspace methods. Our starting point is the Schur complement formulation of the mixed problem. One of the difficulties encountered with the approach we consider is generally associated with the issue of defining boundary conditions for the associated Schur complement problem. To derive suitable boundary conditions for this problem, we first consider an iterative solution method which is based on the direct use of the Schur complement formulation. This development naturally leads us to address the key question of what boundary conditions should be adjoined to the Schur complement preconditioner. We give numerical examples which demonstrate the effectiveness and robustness of the proposed preconditioner. The examples include both two- and three-dimensional problems with boundary conditions of various types.