Approximation of multi-scale elliptic problems using patches of finite elements

The objective of this seminar is to present a new method to solve numerically
elliptic problems such that a better precision on the solution is needed
in certain regions of the domain wherein the equations
are to be solved (C.R.Acad.Sci.Paris, Ser.I 337 (2003) 679--684).
The approximation of this type of problems with multi-scale data can be approached using
various methods. The technique we present uses multiple levels of not necessarily nested
grids. It is a Schwarz type domain decomposition method with complete overlapping.
The proposed algorithm consists in solving the problem on a domain wherein we consider
patches of elements in the regions where we would like to obtain more accuracy.
Thus we calculate successively corrections to the solution in the patches.
The discretizations of the latter are not necessarily conforming.
The method resembles the Fast Adaptive Composite grid method or possibly
a hierarchical method with a mortar method. However it is of much more flexible use
in comparison to the latter.

The motivation for developing such a method is for example founded in air quality management.
Pollution emission sources, and in particular point source plumes, give
rise to models needing careful examination of the space-scale. Getting an accurate
simulation on large scales is linked to a simulation in subregions around the
pollution sources using finer grids. Such a method can straightforwardly be applied
on boundary layer problems through the use of patches in critical regions.

In this talk we present the algorithm and illustrate its efficiency through a model problem.
We compare the speed of convergence on nested and non-nested, structured and unstructured grids.
A spectral analysis of the iteration operator enables us to give a good estimate of the convergence
rate for given grids. It also leads to a numerical method to evaluate the
constant of the Cauchy-Buniakowski-Schwarz inequality in certain cases of approximation spaces.
Finally we illustrate on several examples our \emph{a priori} estimate for
the convergence in the grid-size.