Toward a General Solution Verification Method for Complex PDE Problem with Hands Off Coding

Abstract

The goal of this paper is to present a versatile framework for solution verification of PDEs. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that constructs the best consistent solution from a set of two or three coarse grid solution in the discrete norm of choice. This technique generalizes the Least Square Extrapolation method introduced by one of the author and W.Shyy. We second establish the conditioning number of the problem in a reduced space that approximates the main feature of the numerical solution thanks to a sensitivity analysis. Overall our method produces an a posteriori error estimation in this reduced space of approximation. The key feature of our method is that our construction does not require an internal knowledge of the software neither the source code that produces the solution to be verified. It can be applied in principle as a postprocessing procedure to off the shelf commercial code. We demonstrate the robustness of our method with two steady problems that are respectively an incompressible back step flow test case and a heat transfer problem for a battery. Both test cases are solved by the Finite Element package ADINA. Our error estimate might be ultimately verified with a near by manufactured solution. While our procedure is systematic and requires numerous computation of residuals, one can take advantage of grid computing to get quickly the error estimate.