Applied Mathematic Seminar

Nonlocal calculus, nonlocal balance laws and asymptotically compatible discretizations
Wed, 19 March, 2014 8:00pm

Speaker: Qiang Du, Penn State University

Abstract: Nonlocality is ubiquitous in nature. While partial differential
equations (PDE) have been used as effective models of many physical
processes, nonlocal models and nonlocal balanced laws are also attracting
more and more attentions as possible alternatives to treat anomalous
process and singular behavior. In this talk, we exploit the use of a
recently developed nonlocal vector calculus to study a class of constrained
value problems on bounded domains associated with some nonlocal
balance laws. The nonlocal calculus of variations then offers
striking analogies between nonlocal model and classical local PDE models
as well as the notion of local and nonlocal fluxes. We discuss the
consistency of nonlocal models to local PDE limits as the horizon, which
measures the range of nonlocal interactions, approaches
zero. In addition, we present asymptotically compatible discretizations
that provide convergent approximations in the nonlocal setting with a
nonzero horizon and are also convergent asymptotically to the
local limit as both the horizon and the mesh size are taking to zero.
Such asymptotically compatible discretizations can be more
robust for multiscale problems with varying length scales.
 


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