J. Thomas Beale

  • Professor Emeritus of Mathematics
External address: 231 Physics Bldg, Durham, NC 27708-0320
Internal office address: Box 90320, Durham, NC 27708-0320
Phone: (919) 660-2814
Office Hours: 

by appointment.

Research Areas and Keywords

Analysis
computation of singular and nearly singular integrals, motion of fluid interfaces, equations of incompressible flow, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow
Computational Mathematics
boundary integral methods, computation of singular and nearly singular integrals, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow
PDE & Dynamical Systems
boundary integral methods, motion of fluid interfaces, equations of incompressible flow, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow
Physical Modeling
boundary integral methods, motion of fluid interfaces, equations of incompressible flow, convergence of numerical methods for fluid flow

Here are two recent papers:
J. T. Beale, W. Ying, and J. R. Wilson, A simple method for computing singular or nearly singular integrals on closed surfaces,  Comm. Comput. Phys. 20 (2016), 733-753 or  arxiv.org/abs/1508.00265
J. T. Beale, Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method, SIAM J. Numer. Anal. 53 (2015), 2097-2111 or  arxiv.org/abs/1503.05810

Much of my work has to do with incompressible fluid flow, especially qualitative properties of solutions and behavior of numerical methods, using analytical tools of partial differential equations. My research of the last few years has the dual goals of designing numerical methods for problems with interfaces, especially moving interfaces in fluid flow, and the analysis of errors in computational methods of this type. We have developed a general method for the numerical computation of singular or nearly singular integrals, such as layer potentials on a curve or surface, evaluated at a point on the curve or surface or nearby, in work with M.-C. Lai, A. Layton, S. Tlupova, and W. Ying. After regularizing the integrand, a standard quadrature is used, and corrections are added which are determined analytically. Current work with coworkers is intended to make these methods more practical, especially in three dimensional simulations. Some projects (partly with Anita Layton) concern the design of numerical methods which combine finite difference methods with separate computations on interfaces. We developed a relatively simple approach for computing Navier-Stokes flow with an elastic interface. In analytical work we have derived estimates in maximum norm for elliptic (steady-state) and parabolic (diffusive) partial differential equations. For problems with interfaces, maximum norm estimates are more informative than the usual ones in the L^2 sense. More general estimates were proved by Michael Pruitt in his Ph.D. thesis.

Education & Training
  • Ph.D., Stanford University 1973

  • M.S., Stanford University 1969

  • B.S., California Institute of Technology 1967

Beale, JT. "The existence of cnoidal water waves with surface tension." Journal of Differential Equations 31.2 (1979): 230-263.

Beale, JT. "The existence of solitary water waves." Communications on Pure and Applied Mathematics 30.4 (July 1977): 373-389. Full Text

Beale, JT. "Eigenfunction expansions for objects floating in an open sea." Communications on Pure and Applied Mathematics 30.3 (May 1977): 283-313. Full Text

Beale, JT. "Acoustic Scattering From Locally Reacting Surfaces." Indiana University Mathematics Journal 26.2 (1977): 199-222.

Beale, JT. "Spectral Properties of an Acoustic Boundary Condition." Indiana University Mathematics Journal 25.9 (1976): 895-917.

Beale, JT, and Rosencrans, SI. "Acoustic boundary conditions." Bulletin of the American Mathematical Society 80.6 (November 1, 1974): 1276-1279. Full Text

Beale, JT. "Purely imaginary scattering frequencies for exterior domains." Duke Mathematical Journal 41.3 (September 1974): 607-637. Full Text

Beale, JT. "Scattering frequencies of resonators." Communications on Pure and Applied Mathematics 26.4 (July 1973): 549-563. Full Text

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