In partial fulfillment of the Requirements for the Degree of
PhD
Christophe Picard
will defend his thesis
A posteriori error estimator framework for
partial differential equations
Abstract
The Least Square Extrapolation (LSE) method for solution verification was introduced in 2002. Since, the method had forked into three areas.
The method was extended to the study of stiff elliptic problems. Techniques developed in this new framework allow to have a rigorous upper bound error estimator to predict very fine grid solutions. This work is applicable to the pressure solver in Immersed Boundary Method. But the equations are still steady.
In this PhD, we expanded the method to two more type of problems.
First, LSE method was extended to parabolic equations by using coarse grids solutions that have different meshes in space and time, with minimum overhead on memory.
Finally, we designed a new method that offers a general framework to do solution verification efficiently by processing the underlying set of discrete (non)-linear equations without using a priori information on the approximation theory framework that is applied to solve the PDE. A library was developed to compute the optimized extrapolation using the surface response methodology. Any 3D Navier-Stokes code can be plug in it to compute the optimum without knowledge of the internal structure of the code. This work has evolved by establishing 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. Standard benchmark problems showed that more information can be extracted than by using Richardson Extrapolation.