Description
Speaker: Prof. Jerome Droniou
Affiliation: Monash University, Melbourne
Time: Monday (15-10-18), 2.30 PM-3.30 PM.
Venue: Ramanujan Hall.
Tiitle: What the second Strang lemma and the Aubin-Nitsche 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 non-conforming 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. Ad-hoc
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 Aubin-Nitsche 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
Aubin-Nitsche trick are extremely similar to those appearing when applying
the third Strang lemma; work done in the latter case can therefore be
re-invested when looking for improved estimates in a weaker norm.
I will conclude by briefly presenting applications of the third Strang
lemma and the abstract Aubin-Nitsche trick to discontinuous Galerkin and
Finite Volume methods.