# 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

##### Computational Mathematics

##### PDE & Dynamical Systems

##### Physical Modeling

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.

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.
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Baker, GR, and Beale, JT. "Vortex blob methods applied to interfacial motion." *Journal of Computational Physics* 196.1 (2004): 233-258.
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Beale, JT. "Discretization of Layer Potentials and Numerical Methods for Water Waves (Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics)." *RIMS Kokyuroku* 1234 (October 2001): 18-26.

Beale, JT, and Lai, M-C. "A method for computing nearly singular integrals." *SIAM Journal on Numerical Analysis* 38.6 (2001): 1902-1925.
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Beale, JT. "A convergent boundary integral method for three-dimensional water waves." *Mathematics of Computation* 70.235 (2001): 977-1029.
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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.
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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.