A Least Square Extrapolation Method for the A Posteriori Error Estimate of CFD and Heat Transfer Problem

A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for elliptic operators in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates on coarse grid solutions. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced [Garbey 13 international conference on domain decomposition and Garbey & Shyy JCP 2003] a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces an optimization framework facilitating multi-level extrapolation, improves accuracy and provides a posteriori error estimate. We will illustrate the method with incompressible Navier Stokes flow computations and present a new development of the method for unsteady heat transfer problems.