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.

Office Location: | 231 Physics Bldg, Durham, NC 27708-0320 |

Office Phone: | (919) 660-2814 |

Email Address: | |

Web Page: | http://www.math.duke.edu/cncs/~beale |

**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 and Fluid Mechanics*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. Recently I have developed a general method for the numerical computation of singular integrals, such as layer potentials on a curve or surface, evaluated at a point on or near the layer (partly with M.-C. Lai). A standard quadrature is used for a regularized integral on overlapping coordinate grids with analytical corrections. In work with J. Strain we have applied this approach to Stokes flow with a moving elastic interface. A recent paper with A. Layton gives an analytical explanation, with applications, of the observed gain in order of accuracy in certain finite difference methods for computing boundary value problems with irregular boundaries using only regular grids (A. Mayo's method or the immersed interface method of R. LeVeque and Z. Li). Related projects include computation of unstable fluid interfaces (with G. Baker) and the design of a boundary integral method for 3-D water waves which is numerically stable and proved to converge; the discretization of potentials on a moving surface is a central issue.

**Keywords:**Differential equations, Partial • Fluid mechanics • Fluid-structure interaction • Numerical analysis

**Current Ph.D. Students**

**Representative 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 Navier-Stokes flow with an exact moving boundary using the immersed interface method*, Siam Journal on Numerical Analysis, vol. 53 no. 4 (January, 2015), pp. 2097-2111, Society for Industrial & Applied Mathematics (SIAM), ISSN 0036-1429 [pdf], [doi] [abs] - Tlupova, S; Beale, JT,
*Nearly singular integrals in 3D stokes flow*, Communications in Computational Physics, vol. 14 no. 5 (2013), pp. 1207-1227, Global Science Press, ISSN 1815-2406 [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. 1073-1093, Global Science Press, ISSN 1815-2406 [pdf], [doi] [abs] - Beale, JT,
*Partially implicit motion of a sharp interface in Navier-Stokes flow*, J. Comput. Phys., vol. 231 no. 18 (2012), pp. 6159-6172, 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. 1139-1153, American Institute of Mathematical Sciences (AIMS), ISSN 1531-3492 [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. 2476-2495, Society for Industrial & Applied Mathematics (SIAM), ISSN 0036-1429 [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. 3358-3367, Elsevier BV, ISSN 0021-9991 [pdf], [doi] [abs] - Beale, JT,
*A proof that a discrete delta function is second-order accurate*, Journal of Computational Physics, vol. 227 no. 4 (February, 2008), pp. 2195-2197, Elsevier BV, ISSN 0021-9991 [pdf], [doi] [abs] - Beale, JT; Strain, J,
*Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces*, Journal of Computational Physics, vol. 227 no. 8 (April, 2008), pp. 3896-3920, Elsevier BV, ISSN 0021-9991 [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. 91-119, 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. 233-258, Elsevier BV [pdf], [doi] [abs] - Beale, JT,
*A grid-based boundary integral method for elliptic problems in three dimensions*, Siam Journal on Numerical Analysis, vol. 42 no. 2 (December, 2004), pp. 599-620, Society for Industrial & Applied Mathematics (SIAM), ISSN 0036-1429 [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. 1902-1925, Society for Industrial & Applied Mathematics (SIAM) [ps], [doi] [abs] - Beale, JT,
*A convergent boundary integral method for three-dimensional water waves*, Mathematics of Computation, vol. 70 no. 235 (July, 2001), pp. 977-1029, American Mathematical Society (AMS) [ps], [doi] [abs]

- J. t. Beale, W. YIng, and J. R. Wilson,