UMBC
Location
Mathematics/Psychology : 101
Date & Time
April 13, 2018, 2:00 pm – 3:00 pm
Description
Title:
How to model small holes without meshing them
Speaker: Manil Suri, UMBC
Abstract:
Approximating
PDEs over a domain with small holes can be a challenge, due to two reasons.
First, the mesh can be difficult to construct due to the geometry, requiring human
input, which can be expensive. Second, the solution can be unbounded in the
limit as the maximum diameter d of
the holes tends to zero. This leads to poor finite element convergence even for
d > 0, unless highly graded meshes
are used. Since quantities of engineering interest (e.g. total boundary flux)
can still remain bounded as d tends
to 0, one strategy is to calculate the asymptotic (d=0) limit and use it as an approximation for the desired d > 0 case. We show that this
strategy can be quite inaccurate. Instead, we propose an alternative that
completely dispenses with the meshing of the holes. Rather, our numerical
method combines analytic singularities of the solution with the optimal finite
element approximation of its smooth components. We present theoretical and
numerical results to establish the efficacy of our method, both in the energy
norm and in extracting a representative quantity of interest.
This is joint work with Ivo Babuska (UT Austin) and Ana Maria Soane (US Naval Academy). A paper on this talk can be accessed at this link: https://manilsuri.umbc.edu/files/2014/11/CMAME2017.pdf
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