# 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 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), 568-584 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), 605-626 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), 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.

### Selected Grants

Development and Analysis of Numerical Methods for Fluid Interfaces awarded by National Science Foundation (Principal Investigator). 2013 to 2017

Tlupova, S., and J. T. Beale. “Regularized single and double layer integrals in 3D Stokes flow.” *Journal of Computational Physics*, vol. 386, June 2019, pp. 568–84. *Scopus*, doi:10.1016/j.jcp.2019.02.031.
Full Text

Beale, J. T., and W. Ying. “Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation.” *Numerische Mathematik*, vol. 141, no. 3, Mar. 2019, pp. 605–26. *Scopus*, doi:10.1007/s00211-018-1010-2.
Full Text

Beale, J. T., et al. “A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces.” *Communications in Computational Physics*, vol. 20, no. 3, Sept. 2016, pp. 733–53. *Scopus*, doi:10.4208/cicp.030815.240216a.
Full Text

Beale, J. T. “Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method.” *Siam Journal on Numerical Analysis*, vol. 53, no. 4, Jan. 2015, pp. 2097–111. *Scopus*, doi:10.1137/151003441.
Full Text

Tlupova, S., and J. T. Beale. “Nearly singular integrals in 3D stokes flow.” *Communications in Computational Physics*, vol. 14, no. 5, 2013, pp. 1207–27. *Scival*, doi:10.4208/cicp.020812.080213a.
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Ying, W., and J. T. Beale. “A fast accurate boundary integral method for potentials on closely packed cells.” *Communications in Computational Physics*, vol. 14, no. 4, 2013, pp. 1073–93. *Scival*, doi:10.4208/cicp.210612.240113a.
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Layton, A. T., and J. T. Beale. “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. 1139–53. *Scopus*, doi:10.3934/dcdsb.2012.17.1139.
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Beale, J. T. “Partially implicit motion of a sharp interface in Navier-Stokes flow.” *J. Comput. Phys.*, vol. 231, 2012, pp. 6159–72.

Beale, J. T. “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. 2476–95. *Scopus*, doi:10.1137/080731645.
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Beale, J. T., and A. T. Layton. “A velocity decomposition approach for moving interfaces in viscous fluids.” *Journal of Computational Physics*, vol. 228, no. 9, May 2009, pp. 3358–67. *Scopus*, doi:10.1016/j.jcp.2009.01.023.
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## Pages

Professor J Thomas Beale will retire from the mathematics Department in August after 33 years at Duke. A native of Georgia, he was an undergraduate at Caltech and earned his Ph.D. in mathematics at Stanford with the supervision of Ralph Phillips. He... read more »