Nick Trefethen (Oxford)
Rational approximation of functions with corner singularities
Abstract:
Rational functions have extraordinary power for approximation of functions with branch point boundary singularities. Thanks to the exponential clustering of poles, their convergence is root-exponential, i.e., exponential in the square root of the
degree. This was discovered by Newman in 1964 for approximation
of |x| on [-1,1] but only recently applied to the development
of "lightning solvers" for Laplace, biharmonic, and Helmholtz
problems in planar regions with corners. We will present some of
these fast new numerical methods and then focus on a mathematical
aspect that turns up wherever you look: why do those exponentially-
clustered poles space out super-exponentially near the singularity
in a manner described by a certain square root formula? This is
joint work with Yuji Nakatsukasa and Andre Weideman.