Adaptive domain decomposition for the bound method: Application to the incompressible Navier–Stokes and Energy equations in three space dimensions

Large-scale numerical simulations of the flow and associated transport phenomena governed by the Navier–Stokes and Energy equations are routinely calculated in engineering practice. Nevertheless, the uncertainty due to spatial discretization limits the confidence of practitioners in numerical solutions. An approach to provide information about the accuracy of the quantity of interest is proposed here-in. The novel a posteriori   error estimation technique – the bound method – is based on relaxing Lagrange multipliers that enforces continuity between sub-domains. The method provides fast, efficient, asymptotic but reliable lower and upper bounds to the output of underlying partial differential equations (PDEs). Herein, we highlight the method when applied to outputs of the steady incompressible Navier–Stokes and Energy equations. The bound method in this paper follows the directly equilibrated hybrid-flux approach for the flux calculation between sub-domains and uses the Crouzeix–Raviart (P2+-P1) approximation spaces. To improve the effectiveness of the bound method, an adaptive sub-domain refinement strategy leading to sharper bounds is adopted. A convective heat transfer problem in a series of electronic chip devices is investigated. The novelty of this paper is to present bounds using adaptive domain decomposition for outputs associated to a complex three-dimensional field solution of the Navier–Stokes and Energy equations.