Math @ Duke

J. Thomas Beale, Professor
 Contact Info:
Office Location:  217 Physics  Office Phone:  (919) 6602839  Email Address:   Teaching (Fall 2014):
 MATH 555.01, ORDINARY DIFF EQUATIONS
Synopsis
 Physics 205, MWF 03:05 PM03:55 PM
 Office Hours:
 Wednesday and Friday, 2:00  3:00
 Education:
 B.S., Caltech, 1967
M.S., Stanford University, 1969
Ph.D., Stanford University, 1973
 Specialties:

Analysis
Applied Math
 Research Interests: Partial Differential Equations, Fluid Mechanics, Numerical Methods
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 NavierStokes flow with an elastic interface.
In analytical work we have derived estimates
in maximum norm for elliptic (steadystate) 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.
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Representative Publications
(More Publications)
 S. Tlupova and J. T. Beale, Nearly singular integrals in 3D Stokes flow,
Commun. Comput. Phys., vol. 14
(2013),
pp. 120727 [pdf]
 W. Ying and J. T. Beale, A fast accurate boundary integral method for potentials on closely packed cells,
Commun. Comput. Phys., vol. 14
(2013),
pp. 107393 [pdf]
 J. T. Beale, Partially implicit motion of a sharp interface in NavierStokes flow,
J. Comput. Phys., vol. 231
(2012),
pp. 615972 [pdf]
 A. T. Layton and J. T. Beale, A partially implicit hybrid method for computing interface motion in Stokes flow,
Discrete and Continuous Dynamical Systems B, vol. 17
(2012),
pp. 113953 [pdf]
 J. T. Beale, Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm,
SIAM J. Numer. Anal., vol. 47
(2009),
pp. 247695 [pdf]
 J. T. Beale and A. T. Layton, A velocity decomposition approach for moving interfaces in viscous fluids,
J. Comput. Phys. 228, 335867
(2009) [pdf]
 J. T. Beale, A proof that a discrete delta function is secondorder accurate,
J. Comput. Phys., vol. 227
(2008),
pp. 219597 [pdf]
 J. T. Beale and J. Strain, Locally corrected semiLagrangian methods for Stokes flow with moving elastic interfaces,
J. Comput. Phys., vol. 227
(2008),
pp. 38963920 [pdf]
 J. T. Beale and A. T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces,
Commun. Appl. Math. Comput. Sci., vol. 1
(2006),
pp. 91119 [pdf]
 G. R. Baker and J. T. Beale, Vortex blob methods applied to interfacial motion,
J. Comput. Phys., vol. 196
(2004),
pp. 23358 [pdf]
 J. T. Beale, A gridbased boundary integral method for elliptic problems in three dimensions,
SIAM J. Numer. Anal., vol. 42
(2004),
pp. 599620 [pdf]
 J. T. Beale, M.C. Lai, A Method for Computing Nearly Singular Integrals,
SIAM J. Numer. Anal., 38 (2001), 190225
[ps]
 J. T. Beale, A Convergent Boundary Integral Method for ThreeDimensional Water Waves,
Math. Comp. 70 (2001), 9771029
[ps]
 Recent Grant Support
 Development and Analysis of Numerical Methods for Fluid Interfaces, National Science Foundation, 2013/082016/07.


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

