A code-independent technique for computational veri cation of uid mechanics and heat transfer problems

This paper addresses the challenge of Solution Veri cation (SV) and accuracy assessment for computing complex Partial Di erential Equation (PDE) model. Our goal is to provide a postprocessing software infrastructure that can be connected to any existing numerical simulation software, for example, widely used commercial applications such as ADINA, Ansys, Fluent, Numeca, Star-CD etc... and provide an a posteriori error estimate to their simulation. Important design decisions are based on simulation done with these software. Unfortunately, we know that to verify a numerical solution, that is to provide a quantitative assessment on the numerical accuracy of the solution, is di cult. The problem of accuracy assessment is a necessary step that should be treated between the code veri cation step and the code validation step to complete the global task of providing a reliable virtual experiment tool [1, 2]. Our major goal in this paper is to pursue our work on the design of a new method that o ers a general framework to do solution veri cation e ciently [3, 4, 5]. The standard approach in applied mathematics to handle the problem of solution veri cation is to work on the approximation theory of the PDE. For each speci c PDE problem, the right Finite Element (FE) approximation may provide the correct a posteriori error estimate [6]. Unfortunately, this approach may require a complete rewriting of an existing Computational Fluid Dynamic (CFD) application based on Finite Volume (FV) for example, and lack generality. Usually a posteriori estimators fails if the (nonlinear) PDE solution is sti or if the grid resolution is not adequate. Since grid re nement itself is based on a posteriori estimator, this leads to an obvious problem. Large Reynolds number ow are common in many applications, including turbulence problems. For those applications rigorous solution veri cation may not be achievable by the current state of the art of numerical analysis. The di culty of SV is even greater for complex multi-physic coupling. The general practice in scienti c computing is to simulate PDEs, for which neither applied mathematics, nor numerical analysis, guaranty the result. Because of the time lag between the development of rigorous mathematical tools and the common scienti c computing practice, our goal is to improve existing SV tools such as the convergence index of Roache et Al, and the Richardson Extrapolation (RE) technique [7, 8], that are used daily by practitioner, by something more elaborate and