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Xiaochun Tian : Interface problems with nonlocal diffusion

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, they also come with increased difficulty in numerical analysis with nonlocality involved. In the first part of this talk, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results. In the second part of this talk, I will describe our recent effort in computing a nonlocal interface problem arising from segregation of two species with high competition. One species moves according to the classical diffusion and the other adopts a nonlocal strategy. A novel iterative scheme will be presented that constructs a sequence of supersolutions shown to be convergent to the viscosity solution of the interface problem.

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