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, submitted to Comm. Comput. Phys., 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, to appear in SIAM J. Numer. Anal., 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:||217 Physics Bldg, Durham, NC 27708-0320|
|Office Phone:||(919) 660-2839|
Teaching (Spring 2016):
|B.S.||California Institute of Technology||1967|
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.