## Applied Mathematics Colloquium: Dr. Manil Suri

#### UMBC

Friday, April 13, 2018 · 2 PM - 3 PM

**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