Atlas Mathematical Conference Abstracts || Conferences | Abstracts | for Organizers | About AMCA

5th IMACS Conference on Iterative Methods in Scientific Computing
May 28-31, 2001
Foundation for Research and Technology - Hellas (FORTH)
Heraklion, Crete, Greece

Organizers
Apostolos Hadjidimos, Elias Houstis, Emmanuel Vavalis

View Abstracts
Conference Homepage

Analysis of an interface relaxation method for elliptic differential equations
by
Panagiota Tsompanopoulou
Purdue University, Computer Science Department, West Lafayette, IN 47907, USA
Coauthors: John R. Rice, Emmanuel Vavalis

article The main objective of our paper is to theoretically analyze and experimentally investigate and evaluate a new Interface Relaxation method for solving elliptic differential equations. This method iterates to relax the values on the interfaces by adding to the old ones, a geometrically weighted combination of the normal boundary derivatives of the adjacent subdomains.

Specifically, we consider the following relaxation formula on the interface lines


u(k+1) = u(k) - \rho ( \frac\partialuL(k)\partial\eta + \frac\partialuR(k)\partial\eta )

where k denotes the iteration step, u the approximation of the solution on the interface, and \frac\partialuL\partial\eta, \frac\partialuR\partial\eta the values of the normal derivatives of the computed solution on the two adjacent subdomains. \rho is a relaxation parameter used to accelerate the convergence of the iteration scheme.

This method was first briefly proposed in [2], where its derivation, based on a simple geometric contraction mechanism, can be found together with preliminary 1-dimensional experimental data. A variation of this method has been also considered in [1] where a convergence analysis and numerical data are presented.

Our theoretical analysis is carried out for the 1-dimensional and 2-dimensional Helhmoltz equation with 1-dimensional decompositions. The convergence of the method is proved, and the convergence region and the optimal values for the relaxation parameters involved are determined. Numerical data that confirm our convergence results and show that they hold for more general problems and decompositions are also presented.

This work is part of the speaker's Ph.D. thesis.

[]
M. Mu. Solving composite problems with interface relaxation. SIAM J. Sci. Comput., 1999 (to appear).

[]
J.R. Rice, P. Tsompanopoulou, and E. Vavalis. Review and performance interface relaxation methods for elliptic PDEs. Applied Numerical Mathematics, (2000 to appear).

Author's Home Page

Date received: January 9, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cagm-03.