
J. Thomas Beale, Professor of Mathematics
 Contact Info:
Teaching (Fall 2015):
 MATH 653.01, ELLIPTIC PDE
Synopsis
 Physics 047, TuTh 10:05 AM11:20 AM
 Education:
 B.S., Caltech, 1967
M.S., Stanford University, 1969
Ph.D., Stanford University, 1973
 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 3D water waves which is numerically stable and proved to converge; the discretization of potentials on a moving surface is a central issue.
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Representative Publications
(More Publications)
 J. T. Beale, Uniform error estimates for NavierStokes flow with a moving boundary using the immersed interface method,
SIAM J. Numer. Anal.
(Submitted, 2015) [pdf]
 S. Tlupova and J. T. Beale, Nearly singular integrals in 3D Stokes flow,
Commun. Comput. Phys., vol. 14
(2013),
pp. 120727 [pdf]
 W. Ying and J. T. Beale, A fast accurate boundary integral method for potentials on closely packed cells,
Commun. Comput. Phys., vol. 14
(2013),
pp. 107393 [pdf]
 J. T. Beale, Partially implicit motion of a sharp interface in NavierStokes flow,
J. Comput. Phys., vol. 231
(2012),
pp. 615972 [pdf]
 A. T. Layton and J. T. Beale, A partially implicit hybrid method for computing interface motion in Stokes flow,
Discrete and Continuous Dynamical Systems B, vol. 17
(2012),
pp. 113953 [pdf]
 J. T. Beale, Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm,
SIAM J. Numer. Anal., vol. 47
(2009),
pp. 247695 [pdf]
 J. T. Beale and A. T. Layton, A velocity decomposition approach for moving interfaces in viscous fluids,
J. Comput. Phys. 228, 335867
(2009) [pdf]
 J. T. Beale, A proof that a discrete delta function is secondorder accurate,
J. Comput. Phys., vol. 227
(2008),
pp. 219597 [pdf]
 J. T. Beale and J. Strain, Locally corrected semiLagrangian methods for Stokes flow with moving elastic interfaces,
J. Comput. Phys., vol. 227
(2008),
pp. 38963920 [pdf]
 J. T. Beale and A. T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces,
Commun. Appl. Math. Comput. Sci., vol. 1
(2006),
pp. 91119 [pdf]
 G. R. Baker and J. T. Beale, Vortex blob methods applied to interfacial motion,
J. Comput. Phys., vol. 196
(2004),
pp. 23358 [pdf]
 J. T. Beale, A gridbased boundary integral method for elliptic problems in three dimensions,
SIAM J. Numer. Anal., vol. 42
(2004),
pp. 599620 [pdf]
 J. T. Beale, M.C. Lai, A Method for Computing Nearly Singular Integrals,
SIAM J. Numer. Anal., 38 (2001), 190225
[ps]
 J. T. Beale, A Convergent Boundary Integral Method for ThreeDimensional Water Waves,
Math. Comp. 70 (2001), 9771029
[ps]
