J. Thomas Beale, Professor Emeritus
Here are five recent papers: J. T. Beale, Solving partial differential equations on closed surfaces with planar Cartesian grids, submitted to SIAM J. Sci. Comput., arxiv.org/abs/1908.01796 J. T. Beale and S. Tlupova, Regularized single and double layer integrals in 3D Stokes flow, J. Comput. Phys. 386 (2019), 568584 or arxiv.org/abs/1808.02177 J. T. Beale and W. Ying, Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation, Numer. Math. 141(2019), 605626 or arxiv.org/abs/1803.08532 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), 733753 or arxiv.org/abs/1508.00265 J. T. Beale, Uniform error estimates for NavierStokes flow with an exact moving boundary using the immersed interface method, SIAM J. Numer. Anal. 53 (2015), 20972111 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 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.  Contact Info:
Office Location:  231 Physics Bldg, Durham, NC 277080320  Office Phone:  (919) 6602814  Email Address:    Office Hours:
 by appointment.
 Education:
Ph.D.  Stanford University  1973 
M.S.  Stanford University  1969 
B.S.  California Institute of Technology  1967 
 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.
 Keywords:
Differential equations, Partial • Fluid mechanics • Fluidstructure interaction • Numerical analysis
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Representative Publications
(More Publications)
 J. t. Beale, W. YIng, and J. R. Wilson, A simple method for computing singular or nearly singular integrals on closed surfaces,
Commun. Comput. Phys.
(Submitted, August, 2015) [pdf]
 Beale, JT, Uniform error estimates for NavierStokes flow with an exact moving boundary using the immersed interface method,
Siam Journal on Numerical Analysis, vol. 53 no. 4
(January, 2015),
pp. 20972111, Society for Industrial & Applied Mathematics (SIAM), ISSN 00361429 [pdf], [doi] [abs]
 Tlupova, S; Beale, JT, Nearly singular integrals in 3D stokes flow,
Communications in Computational Physics, vol. 14 no. 5
(2013),
pp. 12071227, Global Science Press, ISSN 18152406 [pdf], [doi] [abs]
 Ying, W; Beale, JT, A fast accurate boundary integral method for potentials on closely packed cells,
Communications in Computational Physics, vol. 14 no. 4
(2013),
pp. 10731093, Global Science Press, ISSN 18152406 [pdf], [doi] [abs]
 Beale, JT, Partially implicit motion of a sharp interface in NavierStokes flow,
J. Comput. Phys., vol. 231 no. 18
(2012),
pp. 61596172, Elsevier BV [pdf], [doi]
 Layton, AT; Beale, JT, A partially implicit hybrid method for computing interface motion in stokes flow,
Discrete and Continuous Dynamical Systems Series B, vol. 17 no. 4
(June, 2012),
pp. 11391153, American Institute of Mathematical Sciences (AIMS), ISSN 15313492 [pdf], [doi] [abs]
 Beale, JT, Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm,
Siam Journal on Numerical Analysis, vol. 47 no. 4
(July, 2009),
pp. 24762495, Society for Industrial & Applied Mathematics (SIAM), ISSN 00361429 [pdf], [doi] [abs]
 Beale, JT; Layton, AT, A velocity decomposition approach for moving interfaces in viscous fluids,
Journal of Computational Physics, vol. 228 no. 9
(May, 2009),
pp. 33583367, Elsevier BV, ISSN 00219991 [pdf], [doi] [abs]
 Beale, JT, A proof that a discrete delta function is secondorder accurate,
Journal of Computational Physics, vol. 227 no. 4
(February, 2008),
pp. 21952197, Elsevier BV, ISSN 00219991 [pdf], [doi] [abs]
 Beale, JT; Strain, J, Locally corrected semiLagrangian methods for Stokes flow with moving elastic interfaces,
Journal of Computational Physics, vol. 227 no. 8
(April, 2008),
pp. 38963920, Elsevier BV, ISSN 00219991 [repository], [doi] [abs]
 Thomas Beale, J; Layton, AT, On the accuracy of finite difference methods for elliptic problems with interfaces,
Communications in Applied Mathematics and Computational Science, vol. 1 no. 1
(January, 2006),
pp. 91119, Mathematical Sciences Publishers [pdf], [doi] [abs]
 Baker, GR; Beale, JT, Vortex blob methods applied to interfacial motion,
Journal of Computational Physics, vol. 196 no. 1
(May, 2004),
pp. 233258, Elsevier BV [pdf], [doi] [abs]
 Beale, JT, A gridbased boundary integral method for elliptic problems in three dimensions,
Siam Journal on Numerical Analysis, vol. 42 no. 2
(December, 2004),
pp. 599620, Society for Industrial & Applied Mathematics (SIAM), ISSN 00361429 [pdf], [doi] [abs]
 Beale, JT; Lai, MC, A method for computing nearly singular integrals,
Siam Journal on Numerical Analysis, vol. 38 no. 6
(December, 2001),
pp. 19021925, Society for Industrial & Applied Mathematics (SIAM) [ps], [doi] [abs]
 Beale, JT, A convergent boundary integral method for threedimensional water waves,
Mathematics of Computation, vol. 70 no. 235
(July, 2001),
pp. 9771029, American Mathematical Society (AMS) [ps], [doi] [abs]
 Recent Grant Support
 Development and Analysis of Numerical Methods for Fluid Interfaces, National Science Foundation, 2013/082017/07.
