The adaptive finite element method (AFEM) is a very successful scheme for the numerical resolution of partial differential equations (PDEs).

In Finite Element simulations, the domain of a PDE is discretised into a large set of small and simple domains (the cells) depending on a size parameter h. Typical shapes that are used for the discretisation are triangles, quadrilaterals, tetrahedrons, or hexahedrons. The solution space is constructed by gluing together simpler finite dimensional spaces, defined on a piecewise manner on each cell, and the original problem is Solved on this simpler, finite dimensional space, transforming the original PDE into an algebraic system of equations. Rigorous analysis of the numerical method allows one to Estimate the discretisation error both a priori (giving global bounds on the total error that depend on the size parameter h), and a posteriori (providing a local distribution of the error on the discretised mesh). AFEM consists of successive loops of the steps Solve→Estimate→Mark→Refine→... to decrease the total discretisation error, by repeating the solution process on a mesh that has been refined on the areas where the a-posteriori analysis has shown that the error is larger.

In this talk, I will introduce and analyse a Smoothed-AFEM, which consists of replacing the intermediate loops of AFEM with the steps Smooth→Estimate→Mark→Refine→.., where instead of solving the discrete problem, I introduce smoothing iterations, that capture the most oscillatory part of the discrete solution. I will explain how this strategy leads to a quasi-optimal mesh sequence construction by saving a lot of computational time compared to the classical AFEM, but still maintaining the same accuracy and order of approximation of the final approximation. I will present some results in the a posteriori error estimation analysis and in the algebraic error estimation, followed by a numerical validation of the strategy.

This is a joint work with Prof.Luca Heltai and Prof.Stefano Giani.