For more information about this meeting, contact Stephanie Zerby, Mark Levi, Jinchao Xu.

Title: | Computing with Singular and Nearly Singular Integrals |

Seminar: | Computational and Applied Mathematics Colloquium |

Speaker: | Tom Beale, Mathematics, Duke University |

Abstract: |

We will describe a relatively 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 smooth, closed curve in 2D or a surface in 3D. Integral formulations are used especially for Stokes flow (viscosity-dominated fluid flow)and in electromagnetics. The present approach can be useful for moving interfaces since the representation of the interface
requires less work than boundary elements. The value of the integral is found by a standard quadrature, with the singularity replaced by a regularized
version. Correction terms are then added for the errors due to
regularization and discretization. These corrections are found by local
analysis near the singularity. The accurate evaluation of a layer potential
at a point near the curve or surface on which it is defined is not routine,
since the integral is nearly singular. For a surface in 3D, integrals are
computed in overlapping coordinate grids on the surface. A quadrature
technique of J. Wilson allows this to be done without explicit knowledge
of the coordinate charts, thus making the approach more practical.
For a boundary value problem, an integral equation can be solved for the
density of the needed potential. For a fluid interface, e.g. in Stokes flow,
the velocity or pressure can be written in terms of layer potentials. Recent
work with W. Ying in 2D shows that the method can accurately handle boundaries
which are close to each other, such as two drops merging, and work in 3D
is in progress. Collaborators include M.-C. Lai, A. Layton, S. Tlupova,
J. Wilson, and W. Ying. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 03 / 15 / 2013 |

Time: | 03:35pm - 04:25pm |