Singular Pertubation

Gudrun had two podcast conversations at the FEniCS18 workshop in
Oxford (21.-23. March 2018). FEniCS is an open source computing
platform for solving partial differential equations with Finite
Element methods.


This is the first of the two episodes from Oxford in 2018.


Roisin Hill works at the National University of Ireland in Galway
on the west coast of Ireland. The university has 19.000 students
and 2.000 staff. Roisin is a PhD student in Numerical Analysis at
the School of Mathematics, Statistics & Applied Mathematics.
Gudrun met her at her poster about Balanced norms and mesh
generation for singularly perturbed reaction-diffusion problems.
This is a collaboration with Niall Madden who is her supervisor
in Galway.


The name of the poster refers to three topics which are
interlinked in their research. Firstly, water flow is modelled as
a singularly perturbed equation in a one-dimensional channel. Due
to the fact that at the fluid does not move at the boundary there
has to be a boundary layer in which the flow properties change.
This might occur very rapidly. So, the second topic is that
depending on the boundary layer the problem is singularly
perturbed and in the limit it is even ill-posed. When solving
this equation numerically, it would be best, to have a fine mesh
at places where the error is large. Roisin uses a posteriori
information to see where the largest errors occur and changes the
mesh accordingly. To choose the best norm for errors is the third
topic in the mix and strongly depends on the type of singularity.


More precisely as their prototypical test case they look for u(x)
as the numerical solution of the problem


for given functions b(x) and f(x). It is singularly perturbed in
the sense that the positive real parameter ε may be arbitrarily
small. If we formally set ε = 0, then it is ill-posed. The
numercial schemes of choice are finite element methods -
implemented in FEniCS with linear and quadratic elements. The
numerical solution and its generalisations to higher-dimensional
problems, and to the closely related convection-diffusion
problem, presents numerous mathematical and computational
challenges, particularly as ε 0. The development of algorithms
for robust solution is the subject of intense mathematical
investigation. Here “robust” means two things:
The algorithm should yield a “reasonable” solution for all
ranges of ε, including resolving any layers present; The
mathematical analysis of the method should be valid for all ranges
of ε.

In order to measure the error, the energy norm sounds like a good
basis - but as ε^2 0 the norm 0 with order ε . They were
looking for an alternative which they found in the literature as
the so-called balanced norm. That remains O(1) as ε 0.
Therefore, it turns out that the balanced norm is indeed a better
basis for error measurement.


After she finished school Roisin became an accountant. She
believed what she was told: if you are good at mathematics,
accountancy is the right career. Later her daughter became ill
and had to be partially schooled at home. This was the moment
when Roisin first encountered applied mathematics and fell in
love with the topic. Inspired by her daughter - who did a degree
in general science specialising in applied mathematics - Roisin
studied mathematics and is a PhD student now (since Sept. 2017).
Her enthusiasm has created impressive results: She won a
prestigious Postgraduate Scholarship from the Irish Research
Council for her four year PhD program.
References

R. Lin, M. Stynes: A balanced finite element method for
singularly perturbed reaction diffusion problems. SIAM J. Numer.
Anal., 50(5):2729–2743, 2012.

T. Linß: Layer-adapted meshes for
reaction-convection-diffusion problems, volume 1985 of Lecture
Notes in Mathematics. Springer-Verlag, Berlin, 2010.

H.-G. Roos, M. Stynes, L. Tobiska: Robust Numerical Methods
for Singularly Perturbed Differential Equations, volume 24 of
Springer Series in Computational Mathematics. Springer-Verlag,
Berlin, 2nd edition, 2008.

Podcasts

M. E. Rognes: Cerebral Fluid Flow, Gespräch mit G. Thäter im
Modellansatz Podcast, Folge 134, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2017.