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Discretization of the fractional Laplacian using finite element methods and a posteriori error estimation.

par Petit Valentin - publié le

Vendredi 16 octobre 2020
Raphaël Bulle
(University of Luxembourg, Université de Bourgogne Franche-Comté)

Scientists are constantly developing more sophisticated mathematical models in order to accurately describe more and more complex physical systems. In this quest for accurate models, fractional calculus has gain interest in the last decades.
Fractional models are capable of describing non-local behaviors while keeping a relatively low number of parameters. However, the main strength of fractional models is also their major drawback : their non-locality makes them very challenging to deal with numerically since it can require the solve of massive non-sparse linear systems.
In this talk I will introduce the method I am currently working on that provides a way to discretize the fractional Laplacian operator using standard finite element methods. I will first give a brief introduction to finite element methods and a posteriori error estimation and then describe few methods to discretize the fractional Laplacian. Finally, I will show some numerical results to illustrate these methods and mention some of the mathematical challenges to be addressed in order to prove their efficiency.