The authors study optimization-based domain decomposition algorithm for Navier-Stokes equations. At each iteration and in each subdomain, the method requires to solve a given PDE and the related adjoint PDE. The adjoint equation is linear and can be solved at very little cost compared to the given nonlinear PDE. The authors formulate domain decomposition method for steady Navier-Stokes equations in two space dimensions. Generalized stress conditions are imposed on the interfaces between subdomains as a boundary condition, and the functionals on the interfaces between subdomains are defined based on the \(L^2\) norm of the difference between velocity fields from adjacent subdomains and a square of averaged pressure on the entire domain. The authors also present penalized version of the functionals. Weak formulation of the problem is given, and the existence of optimal solution is proved. The authors use the Lagrange multiplier rule to convert the constrained optimization problem into unconstrained minimization problem, prove the existence of nonzero Lagrange multipliers, and obtain the optimality system.
Finite element method is used for derivation of the corresponding discrete problem. The authors give the convergence of the solutions of discretized problems, give error estimates, and consider gradient and nonlinear least square methods for the solution of discretized equations. Some implementation algorithms are given for a test problem (rectangle divided into two subdomains).