"On the accurate solution of PDEs on domains with corners, with applications in microfluidics"
Mar 24, 2021 Schedule:
- Virtual Tea Time
- 03:00 to 03:30 PM Eastern Time (US and Canada)
- Virtual Colloquium
- 03:30 to 04:30 PM Eastern Time (US and Canada)
Boundary integral equations (BIEs) are a useful tool for solving a particular class of partial differential equations (PDEs). Though limited, this class contains common equations from many branches of physics, including fluid dynamics, electromagnetics, acoustics, and solid mechanics, among others. This talk begins with an overview of BIEs in general, and gives their pros and cons compared to other, more commonly used, methods. Briefly, some of the advantages are: 1) extremely high accuracy, 2) reduced computational cost, and 3) relative ease of handling complicated, unbounded, or moving geometries. Some of the disadvantages are: 1) non-applicability to general PDEs, 2) need for highly specialized tools, and 3) strict assumptions on data regularity. The main part of this talk serves to demonstrate a possible approach to mitigate the third disadvantage. Accurately evaluating the solution to a PDE near a corner is a difficult task for most numerical methods. Compared to other methods however, the situation is in general worse for BIEs because singularities occurring at a corner can introduce errors that pollute the solution globally. A compression technique is adopted to circumvent this problem that demonstrates how BIEs can be used to achieve very high accuracy on such problems without introducing additional unknowns. The talk concludes with a model of viscous drops, and we demonstrate how BIEs allow us to model drop movement near sharp interfaces with extremely high accuracy.