An Efficient Spectral Method for the Inextensible Immersed Interface in Incompressible Flows (Communications in Computational Physics, 2018)
In this paper, we study the equation system governing the movement ofan immersed interface in incompressible fluid flows, and propose an efficient methodfor its numerical solution. The particularity of the current model is the inextensibility constraint imposed on the interface. We are interested in constructing a suitablevariational formulation associated to this problem and the well-posedness of the weakproblem. The significance of this variational formulation is that both the inextensibility of the interface and fluid incompressibility are strictly satisfied, and the well-posedness of the associated weak problem is rigorously proved. To the best of theauthors' knowledge, no other models can be claimed to posses these properties. Infact our new formulation renders the inextensibility and the incompressibility constraints into a unique saddle point problem. Then, based on the proposed variationalframework, we design an efficient spectral method for numerical approximations ofthe weak solution. The main contribution of this work are threefold: 1) a variationalframework for the weak solutions of the immersed interface/incompressible equationsand rigorous proof of the well-posedness of the weak problem; 2) a spectral methodfor solving the weak problem, together with a detailed stability analysis for the numerical solutions; 3) efficient implementation technique for the proposed method andsome numerical experiments carried out to confirm the theoretical claims.
Keywords: Spectral method; fluid-structure interaction; inextensible immersed interface; weak problem; well-posedness.