Toward a General Solution Verification Method for Complex PDE problem with Hands off Coding

This paper addresses the challenge of solution verification and accuracy assessment for computing complex Partial Differential Equation (PDE) model. Our main target applications are bio-heat transfer and blood flow simulation problems. However our long term goal is to provide a postprocessing package that can be attached to any existing numerical simulation package, for example widely used commercial codes such as ADINA, Ansys, Fluent, Star-CD etc... and provide an a posteriori error estimate to their simulation. Important design decision are based on simulation done with these softwares. Unfortunately we know that to verify a numerical solution and provide a quantitative assessment on the numerical accuracy of the solution is extremely difficult. The problem of accuracy assessment is a necessary step that comes after the code verification step and before the code validation step to complete the global task of providing a reliable virtual experiment tool. Our major goal in this paper is to pursue our work on the design of a new method that offer a general framework to do solution verification efficiently. The standard approach in applied mathematics to handle the problem of solution verification is to work on the approximation theory of the PDE. For each specific PDE problem, the right Finite Element (FE) approximation may provide the correct a posteriori error estimate. Unfortunately this approach may require a complete rewriting of an existing CFD code based on Finite Volume (FV) for example and lack generality. While a posteriori estimates in FE can be rigorous and has provided a number of interesting numerical analysis theory, its practical value is often questioned by practitioners. Usually such a posteriori estimators fails if the PDE solution is stiff or if the grid resolution is not adequate. Since grid refinement itself is based on a posteriori estimator, this pose an obvious problem. Large Reynolds number flow are common in many applications, not to mention turbulence problems. For those applications solution verification may not be achievable by the current state of the art of numerical analysis. If the PDE problem gets complicated, for example in the presence of fluid-structure interaction, or other complex multi-physic coupling, one faces the same problem. The general practice in scientific computing is to simulate PDEs for which applied mathematics neither numerical analysis guaranty the result. There is no completed analytical work on the 3D Navier Stokes equation that is a century old! But this model is used every day in fluid flow simulation to design air plane, cars, etc... Because of this time lag between the development of rigorous mathematical tools and the common scientific 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, that are used daily by practitioner, by something (far) more elaborate and reliable that can take both advantage of existing a posteriori estimators when they are available, and new distributed computing tools since SV is computer intensive. Our method relies on four main ideas that are (1) the embedding of the problem of error estimation into an optimum design framework that can extract the best information from