Asymptotic Analysis and Domain Decomposition

Marc Garbey
Centre de Calcul Scientifique Parallel
Université Claude Bernard - Lyon I
Lyon, France

We are interested in the development and application of asymptotic methods for the numerical solution of boundary-value problems with critical parameters -- that is, parameters that determine the nature of the solution in some critical way -- for example, in fluid flow (viscosity), combustion (Lewis number), and superconductivity (Ginzburg-Landau parameter). The solution of these problems may remain smooth over a wide range of parameter values, but as the parameters approach critical values, complicated patterns may emerge. The region over which the solution extends may take on the appearance of a patchwork of subregions; on each subregion, the solution is smooth, but between subregions the solution undergoes dramatic changes over very short distances. Shock layers and boundary layers in fluid flow are a visible manifestation of this type of behavior.

Boundary-value problems with critical parameters pose some of the most challenging problems in computational science, and much effort is being spent on developing new techniques for their numerical solution. Some of the most useful techniques, in particular on parallel computing architectures, are based on domain decomposition, where one partitions the domain into subdomains, approximates the solution on each subdomain, and assembles these solutions to obtain an approximate solution on the entire domain. Many criteria, involving considerations from linear algebra to computer architecture, go into the design of a useful domain decomposition method. Our aim is to explore the use of asymptotic methods.

Asymptotic analysis, in particular singular perturbation theory, is the study of boundary-value problems involving critical parameters. It provides a methodology to identify and characterize boundary layers, transition layers, and initial layers; hence, our idea to use asymptotic methods in the design of domain decomposition algorithms.

The figures, taken from Ref. [1] below, show the solution of a singularly perturbed turning-point problem on the rectangle (-2,2) × (-1,1). The solution has a transition layer along a piecewise linear curve through the origin. The domain was decomposed into four overlapping subdomains (the regular subdomain, two transition layers, and a corner layer covering a neighborhood of the origin), and computed with a Schwarz alternating procedure and an iterative procedure on each subdomain.

The following recent publications are available on the web:

1. M. Garbey and H. G. Kaper, “Heterogeneous Domain Decomposition for Singularly Perturbed Elliptic Boundary Value Problems,” SIAM J. Numer. Anal. 34 (1997), 1513-1544
2. M. Garbey and H. G. Kaper, “Asymptotic-Numerical Study of Supersensitivity for Generalized Burgers' Equation,” submitted for publication. Preprint ANL/MCS-P721-0798

Contact:

Hans G. Kaper, MCS Division
Argonne National Laboratory
Argonne, Illinois 60439
E-mail: kaper@mcs.anl.gov
(630) 252-7160

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Last update: October 21, 1998 (HGK)