Venue: Numerical methods and error analysis for singular integrals and moving interfaces in fluids

Speaker: J. Thomas Beale （Duke University, US）

Time: 27 February 2012,15:00-16:00

Location: Room 201, Sir Run Run Shaw Business Adminisration Building, Yuquan Campus

Abstract: 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.

 Biography: www.math.duke.edu/faculty/beale