Anne Greenbaum.

Laplace's equation and the Dirichlet-Neumann map in multiply connected domains online

. (page 1 of 2)
Online LibraryAnne GreenbaumLaplace's equation and the Dirichlet-Neumann map in multiply connected domains → online text (page 1 of 2)
Font size
QR-code for this ebook


DOE/ER/25053-5
UC-32



Courant Institute of
Mathematical Sciences



Laplace's Equation and the Dirichlet-Neumann Map
in Multiply Connected Domains

Anne Greenbaum, Leslie Greengard and
Geoffrey B. McFadden



iupported by U.S. Department of Energy
jrant No. DE-FGO2-88ER25053



m -H S o vlathematics and Computers

O (l» 4_l g d) '

uT) c CO a c Vlarch 1991

tN c 3 a; c
^ < CPS o

;\ s w ,
o (a H 1-1 (C Q E

S O i-l




NEW YORK UNIVERSITY



UNCLASSIFIED

DOE/ER/25053-5

UC-32

Mathematics and Computers



Courant Institute of Mathematical Sciences
New York University



Laplace's Equation and the Dirichlet-Neumann Map
in Multiply Connected Domains

A. Greenbaum, L. Greengard, and G. B. McFadden
March 1991



The first two authors were supported by U. S. Department of Energy Grant DE-FG02-
8SER25053. The third author is at the Compiiting and Applied Mathematics Laboratory,
National Institute of Standards and Technology. He was supported by the NASA Micro-
gravity Science and Applications Program, and the DARPA Applied and Computational
Mathematics Program.

UNCLASSIFIED



DISCLAEVIER



This report was prepared as an account of work sponsored by an agency of the
United States Government. Neither the United States Government nor any agency
thereof, nor any of their employees, makes any warranty, express or implied, or as-
sumes any legal liability or responsibihty for the accuracy, completeness, or useful-
ness of any information, apparatus, product, or process disclosed, or represents that
its use would not infringe privately owned rights. Reference herein to any specific
commerciaJ product, process, or service by trade name, trademark, manufacturer,
or otherwise, does not necessarily constitute or imply its endorsement, recommen-
dation, or favoring by the United States Government or any agency thereof. The
views and opinions of authors expressed herein do not necessarily state or reflect
those of the United States Government or any agency thereof.



Printed in U.S.A.

Available from

National Technical Information Service

U.S. Department of Commerce

5285 Port Royal Road

Springfield, VA 22161



11



Table of Contents

Page
Abstract 1

1. Introduction 2

2. Siniply Connected Domains 3

3. Multiply Connected Domains 5

3.1 A direct formulation 6

3.2 The exterior problem ' 7

4. Formulation of Numerical Method 8

4.1 Solution of the Discrete System 10

5. The Dirichlet-Neumann Map and the Neumann Problem 12

5.1 The Neumann Problem 13

6. Numerical Results 14

7. Conclusions 17
References 18
Figures 21



111



Laplace's Equation and the

Dirichlet-Neumann Map in Multiply

Connected Domains

Anne Greenbaum* Leslie Greengard^

Geoffrey B. McFadden*

March, 1991

Abstract

A variety of problems in material science and fluid dynamics require
the solution of Laplace's equation in multiply connected domains. In-
tegral equation methods are natural candidates for such problems, since
they discretize the boundary alone, require no special effort for free bound-
aries, and achieve superalgebraic convergence rates on sufficiently smooth
domains in two space dimensions, regardless of shape. Current integral
equation methods for the Dirichlet problem, however, require the solution
of A/ independent problems of dimension A', where M is the number of
boundary components and A' is the total number of points in the dis-
cretization. In this paper, we present a new boundary integral equation
approach, valid for both interior and exterior problems, which requires
the solution of a single linear system of dimension A' + M. We solve this
system by making use of an iterative method (GMRES) combined willi
the fast multipole method for the rapid calculation of the necessary matrix
vector products. For a two-dimensional system with 200 components and
100 points on each boundary, we gain a speedup of a factor of 100 from
the new analytic formulation and a factor of 50 from the fast multipole
method. The resulting scheme brings large scale calculations in extremely
complex domains within practical reach.



'Courant Institute of Mathematical Sciences, New York University, New York. New York
10012. The work of this author was supported by the Applied Mathematical Sciences F'rogram
of the U.S. Department of Energy under Contract DEFGO288ER25053.

'Courant Institute of Mathematical Sciences, New York University, New Nork, New York
10012. The work of this author was supported by the Applied Mathematical Sciences Program
of the U.S. Department of Energy under Contract DEFGO288ER25053.

'Computing and Applied Mathematics Laboratory, National Institute of Staiidrti


1

Online LibraryAnne GreenbaumLaplace's equation and the Dirichlet-Neumann map in multiply connected domains → online text (page 1 of 2)