Integral equations

Thursday, March 15, 2018 - 9:00am - 9:50am
Shravan Veerapaneni (University of Michigan)
We present a boundary integral equation method, developed recently, for simulating the coupled electro- and hydro-dynamics of vesicle suspensions subjected to external flow and electric fields. The dynamics of a vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response.
Monday, August 2, 2010 - 11:00am - 12:00pm
Jacob White (Massachusetts Institute of Technology)
Joint work with H. Reid, L. Zhang, C. Coelho, Z. Zhu, T. Klemas, and S. Johnson.

Despite decades of zealous effort, there are remarkably few
applications where fast integral equation solvers dominate. To help
explain why, we will describe effective strategies, assess performance,
and present remaining challenges associated with applying fast solvers
to problems in interconnect model extraction, biomolecular electrostatics,
bodies in microflows, nanophotonics, and Casimir force calculation.

Monday, August 2, 2010 - 2:30pm - 3:00pm
Mary-Catherine Kropinski (Simon Fraser University)
We present an efficient integral equation approach to solve the heat equation in a two-dimensional, multiply connected domain, and with Dirichlet boundary conditions. Instead of using integral equations based on the heat kernel, we take the approach of discretizing in time, first. This leads to a non-homogeneous modified Helmholtz equation that is solved at each time step. The solution to this equation is formulated as a volume potential plus a double layer potential. The volume potential is evaluated using a fast multipole-accelerated solver.
Thursday, August 5, 2010 - 9:00am - 9:30am
Johan Helsing (Lund University)
First I discuss low-threshold stagnation problems in the GMRES
iterative solver. I show how to alleviate them in certain situations.

Then I turn to the main topic of the talk – a method to enhance the
efficiency of integral equation based schemes for elliptic PDEs on
domains with corners, multi-wedge points, and mixed boundary
conditions. The key ingredients are a block-diagonal inverse
preconditioner 'R' and a fast recursion, 'i=1,...,n', where step 'i'
inverts and compresses contributions to 'R' from the outermost

Thursday, August 5, 2010 - 9:30am - 10:00am
James Bremer (University of California)
We describe a collection of techniques which allow
for the fast and accurate solution of boundary integral
equations on two-dimensional domains whose boundaries
have corner points. Our approach has two key advantages
over existing and recently suggested schemes: (1) it
does not require a prior analytic estimates for solutions,
and (2) many aspects of the scheme generalize readily
to singular three-dimensional domains.
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