2:00pm 
[2:30pm] Prof. Jerome Droniou
 Description:
 Speaker: Prof. Jerome Droniou
Affiliation: Monash University, Melbourne
Time: Monday (151018), 2.30 PM3.30 PM.
Venue: Ramanujan Hall.
Tiitle: What the second Strang lemma and the AubinNitsche trick should be
Abstract: The second Strang lemma gives an error estimate for linear
problems written in variational formulation, such as elliptic equations.
It covers both conforming and nonconforming methods, it is widely spread
in the finite element community, and usually considered as the starting
point of any convergence analysis.
For all its potency, it has a number of limitations which prevents its
direct application to other popular methods, such as dG methods, Virtual
Element Methods, Hybrid High Order schemes, Mimetic Methods, etc. Adhoc
adaptations can be found for some of these methods, but no general `second
Strang lemma' has been developed so far in a framework that covers all
these schemes, and others, at once.
In this talk, I will present a `third Strang lemma' that is applicable to
any discretisation of linear variational problems. The main idea to
develop a framework that goes beyond FEM and covers schemes written in a
fully discrete form is to estimate, in a discrete energy norm, the
difference between the solution to the scheme and some interpolant of the
continuous solution. I will show that this third Strang lemma is much
simpler to prove, and use, than the second Strang lemma. It also enables
us to define a clear notion of consistency, including for schemes for
which such a notion was not clearly defined so far, and for which the Lax
principle `stability + consistency implies convergences' holds.
I will also extend the analysis to the AubinNitsche trick, presenting a
generalisation of this trick that covers fully discrete schemes and
provides improved error estimates in a weaker norm than the discrete
energy norm. We will see that the terms to estimate when applying this
AubinNitsche trick are extremely similar to those appearing when applying
the third Strang lemma; work done in the latter case can therefore be
reinvested when looking for improved estimates in a weaker norm.
I will conclude by briefly presenting applications of the third Strang
lemma and the abstract AubinNitsche trick to discontinuous Galerkin and
Finite Volume methods.

