Past Event: Oden Institute Seminar

On the Solution of Elliptic Partial Differential Equations on Regions with Corners

Vladimir Rokhlin, Professor, Computer Science and Mathematics; Director of Graduate Studies in Computer Science, Yale University.

Abstract

Solution of elliptic partial differential equations on regions with non-smooth boundaries (edges, corners, etc.) is a notoriously refractory problem, especially when high accuracy is desired. In this talk, I observe that when the problems are formulated as boundary integral equations of classical potential theory, the solutions (of the integral equations) in the vicinity of corners can be represented by a series of elementary functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of discretization schemes displaying convergence of arbitrarily high order. For most practical purposes, the loss of accuracy associated with the presence of corners (edges, etc.) disappears entirely. The results are illustrated by a number of numerical examples. Bio: Vladimir Rokhlin is Arthur K. Watson Professor Computer Science and Mathematics; Director of Graduate Studies in Computer Science at Yale University. His research interests include fast deterministic and randomized algorithms of computational mathematics, numerical scattering theory, partial differential equations, integral equations, quadrature formulae, numerical harmonic analysis, numerical linear algebra, special functions, “fast” algorithms of numerical linear algebra. Rokhlin is a member of the National Academy of Sciences; Member of the National Academy of Engineering; recipient of the 2001 Leroy P. Steele Prize for a Seminal contribution to Research; recipient of the 2001 Rice University Distinguished Alumnus Award; recipient of the 2006 Institute of Electrical and Electronics Engineers (IEEE) Honorary Membership; 2009 SIAM Fellow; recipient of the 2011 Maxwell Prize from the ICIAM; recipient of the 2014 William Benter Prize. Representative Publications: •“A Fast Algorithm for Particle Simulations,” with L. Greengard, Journal of Computational Physics, 73(1) : 325 (1987). •“Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions,” Applied and Computational Harmonic Analysis 1:82-93, 1993. •“Randomized Algorithms for the Low-Rank Approximation of Matrices,” with E. Liberty, F. Woolfe, P.G. Martinsson, and M. Tygert, PNAS, V. 104, No. 51, pp. 20167-20172, 2007.