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.

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

**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 (2015), pp. 2097-2111, ISSN 0036-1429 [pdf], [doi] - Tlupova, S; Beale, JT,
*Nearly singular integrals in 3D stokes flow*, Communications in computational physics, vol. 14 no. 5 (2013), pp. 1207-1227, 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, ISSN 1815-2406 [pdf], [doi] [abs] - Beale, JT,
*Partially implicit motion of a sharp interface in Navier-Stokes flow*, J. Comput. Phys., vol. 231 (2012), pp. 6159-6172 [pdf] - 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 (2012), pp. 1139-1153, 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 (2009), pp. 2476-2495, 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 (2009), pp. 3358-3367, 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 (2008), pp. 2195-2197, 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 (2008), pp. 3896-3920, ISSN 0021-9991 [repository], [doi] [abs] - Beale, JT; Layton, AT,
*On the accuracy of finite difference methods for elliptic problems with interfaces*, Commun. Appl. Math. Comput. Sci., vol. 1 (2006), pp. 91-119 [pdf] - Baker, GR; Beale, JT,
*Vortex blob methods applied to interfacial motion*, Journal of Computational Physics, vol. 196 no. 1 (2004), pp. 233-258 [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 (2004), pp. 599-620, ISSN 0036-1429 [pdf], [doi] [abs] - Beale, JT; Lai, M-C,
*A method for computing nearly singular integrals*, SIAM Journal on Numerical Analysis, vol. 38 no. 6 (2001), pp. 1902-1925 [ps], [doi] [abs] - Beale, JT,
*A convergent boundary integral method for three-dimensional water waves*, Mathematics of Computation, vol. 70 no. 235 (2001), pp. 977-1029 [ps], [doi] [abs]

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