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, and Strain, J. "Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces." Journal of Computational Physics 227.8 (2008): 3896-3920. Full Text Open Access Copy

Beale, JT, and Layton, AT. "On the accuracy of finite difference methods for elliptic problems with interfaces." Commun. Appl. Math. Comput. Sci. 1 (2006): 91-119. (Academic Article)

Beale, JT. "A grid-based boundary integral method for elliptic problems in three dimensions." SIAM Journal on Numerical Analysis 42.2 (2004): 599-620. Full Text

Baker, GR, and Beale, JT. "Vortex blob methods applied to interfacial motion." Journal of Computational Physics 196.1 (2004): 233-258. Full Text

Beale, JT, and Lai, M-C. "A method for computing nearly singular integrals." SIAM Journal on Numerical Analysis 38.6 (2001): 1902-1925. Full Text

Beale, JT. "A convergent boundary integral method for three-dimensional water waves." Mathematics of Computation 70.235 (2001): 977-1029. Full Text

Lifschitz, A, Suters, WH, and Beale, JT. "The onset of instability in exact vortex rings with swirl." Journal of Computational Physics 129.1 (1996): 8-29. Full Text

Beale, JT, Hou, TY, and Lowengrub, J. "Convergence of a boundary integral method for water waves." SIAM Journal on Numerical Analysis 33.5 (1996): 1797-1843.

Beale, JT, Hou, TY, and Lowengrub, J. "Stability of boundary integral methods for water waves." AMS-IMS-SIAM Joint Summer Research Conference (1996): 241-245.

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