J. Thomas Beale, Duke University
601 Pao Yue-Kong Library
We will describe three related projects: (1) In work with A. Layton we have designed a second-order accurate numerical method for a moving elastic interface in a viscous fluid governed by the Navier-Stokes equations. The velocity is decomposed as the sum of the Stokes velocity and a more regular remainder. The two parts can be computed in different ways. The method can allow partially implicit motion of the interface to increase the time step.(2) We have developed a simple, direct approach to computing a singular or nearly singular integral, such as a harmonic function given by a single or double layer potential on a curve in R^2 or a surface in R^3, evaluated at a point near the curve or surface. The value is found by regularizing the singularity, using a standard quadrature, and then adding correction terms which are found by local analysis near the singularity. We have proved that the discrete version of the integral equation for a boundary value problem converges to the correct solution.(3) We have proved estimates for discrete versions of elliptic and parabolic problems in maximum norm, with a gain of regularity similar to that for the exact equations. These estimates are related to the accuracy of numerical methods for interface problems.