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Langley Research Center .... behavior of the solution at infinity (a radiation condition). ..... impractical since the associated matrix, which we still call Lsp, is full.
NASA ContractorReport 172380 ICASE REPORT NO. 84-21

NASA-CR-172380 19840021491

ICASE SPECTRAL ELLIPTIC

METHODS FOR EXTERIOR PROBLEMS

C. Canuto S. I. Harlharan L. Lu stman

Contracts

Nos. NASI-17070,

NASI-17130

June 1984

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING NASA Langley Research Center, Hampton, Virginia 23665 Operated

by the Universities

Space

Research

Association

LIBRARY 00PY National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23665

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1984

LANGLEYRESEARCH CENTER LIBRARY,NASA I-LA_PIOL% Y!.RGINI_'

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ISSUE 19 PAGE 3882 !CASE-84-21 MAS 1.25;172380

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CATEGORY 64

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NAS1~17878 NASt-t7l UNCLASSIFIED DOCUMENT Spectral methods for extErior elliptic problems TLSP; Floal R~port A(CANUTO~ C.; B/HARIHARAN~ s. I. C!LUSTMAN~ L. PRH~ A/(CNR~ Pav1a~ I tY); C/(Svstems and Applied Scienc~s Corp., Hampton, Va.) Natlonal !cs and Space Admlntstratloo. Langley Research C€nter~ HamPton} Va. AVAIL. IS SAP~ He A03/MF A0l ;*BOUNDARIES;*ELLIPTIC DIFFERENTIAL EQUATIONS;*PROHLEM SQLVING;*SPECTRUM

Hr~HL \ISIS

I

COORDINATES;

r.1_ A.

DOMA

FAST

IER TRANSFORMATIONS/ INFINITY; ITERATION

---------------------------------------------- ....'-----------

SPECTRAL

METHODS

FOR EXTERIOR

ELLIPTIC

PROBLEMS

C. Canuto Instituto dl Analisi Numerlca del C.N.R. C.C. Alberto, 5, 27100 Pavia, Italy

S. I. Harlharan Universityof TennesseeSpace Institute,Tullahoma,TN 37388 and Institutefor ComputerApplicationsin Scienceand Engineering NASA Langley Research Center, Hampton, VA 23665 L. Lustman Systems and Applied 17 Research Road,

Sciences Hampton,

Corporation VA 23666

Abstract

This

paper

problems finite

in

deals

two

element

methods,

is

introduce

a spectral

"infinite

presented

spectral

dimensions.

solutions

the

with

limited

order"

by

it

is

in

interior

of

the

at infinity, scheme.

accuracy

throughout

which

elliptic

difference

of

farfield

the

is compatible results

Although

our analysis

or

numerical

conditions.

Computational

attainable.

the paper,

exterior

finite

accuracy

the numerical

spectral

for

conventional

that

treatment

the spectral

with a simple Laplace problem complex and general cases.

the

found

the order

boundary

to demonstrate

As

approximations

we

We with are deal

covers more

Research for the first author was supported by NASA Contract No. NASI17070 and for the second author by NASA Contract No. NASI-17130 while both were in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665. *Currently

on leave at the University

i

of California,

Los Angeles.

1. INTRODUCTION

In this paper we address some questions concerning applications of spectral methods to elliptic equations in exterior domains.

Our

goal is to investigate if one can obtain spectral accuracy in this class of problems. It is known through other methods that the accuracy of numerical solutions is governed by the order of the farfield conditions in addition to the order of the scheme itself.

As an example for

finite element methods see Bayliss et al. [BGT]. On the other hand boundary integral equations are very effective for exterior problems. They use appropriate

Green's functions which automatically

care of the farfield behavior

take

(see [HMG] and [GW]). However, they

are limited by the need to use explicitly known kernels and thus cannot treat variable coefficient problems.

We believe spectral methods dress these difficulties.

are useful alternative

ways to ad-

Even more than in other numerical proce-

dures, the formulation of the farfield boundary conditions is an es, sential issue: a poor radiation condition may waste the high precision of spectral methods.

A quite natural way of treating this problem

within a spectral context is presented here, and forms the crucial part of this paper.

The basic problem to be treated here is as follows: let D be a simply connected bounded

domain in R _ , whose smooth boundary

will be denoted by F and whose exterior region by _ = R 2 - D.

We

want to solve the problem Lu--O

(1.1)

in f_

u=g

on r

Boo(u) 0

r=

where L is a second order uniformly _q ,g is a smooth

+ y2 oo, elliptic differential

data on 1" and the last condition

behavior of the solution at infinity (a radiation

operator

represents

in the

condition).

As an example of a physical problem described by this formulation, one can think of D as a conductor

in the field of a line source

(located at x_= _-o). Then L is the Laplacian operator electrostatic

potential,

log r. Another

g(z_) = - log]

A, u is the

z_- x_o I and Boo(u)

example is given by the incompressible,

= u-

irrotational

flow around a body D: again L = A, u is now the velocity potential, g(_) = 0 and Boo(u) - u - Uoz - r stream velocity in the x-direction (1.1) is the simplest treatment

log r, where [To is the main

and r is the circulation.

model one can consider,

Although

we will see that the

by spectral methods is quite general. Even for eigenvalue

problems - such as the Helmholtz

equation, which is not solved here

- the numerical

remains applicable.

farfield algorithm

When the elliptic equation is discretized, main is obtained

by placing an artificial boundary

the body at a finite distance.

condition

B(u)=o on roo,

do-

too surrounding

The radiation condition

(1.1) must be replaced by a boundary

(1.2}

the computational

at infinity in

3

which makes the problem mathematically radiation

condition at infinity.

local (or differential)

Such farfield conditions

type (see for instance

of global (or integro-differential)type satisfied

they introduce

exactly

[BGT], [G], [KM]), or

by approximately

by the physical solution

similar con-

on rc_ .

Hence

an error on the numerical solution, which in principle

should be comparable ical scheme.

may be of

{see, e.g., [ADK], [FM], [M],

[MCM]). They are usually obtained ditions

well-posed and mimics the

with the discretization

error of the numer-

The higher the order of the scheme, the higher the

"order" of the farfield radiation produce discretization

condition.

Since spectral

methods

errors which decay faster than algebraically in

the mesh-size, the farfield condition to be prescribed in conjunction with such methods must be particularly

accurate to preserve the high

precision of the interior scheme. The radiation

condition

stringent requirement,

we present here meets optimally

because its error is just the truncation

this error.

Our farfield condition is somewhat analogous to the global boundary condition

used in [MGM], but it takes most advantage

implemented

from being

in a spectral context, both in terms of computational

efficiency and accuracy. ment of the radiation on the solution.

Numerical

evidence shows that our treat-

condition produces

The plan of the paper is as follows.

overall spectral

accuracy

Section 2 is devoted to

the discussion of the farfield conditions on the artificial boundary. Section 3 we describe a pseudospectral

In

algorithm for solving problem

(1.1). Finally, Section 4 contains the results of some numerical tests.

4

2. AN INFINITE-ORDER

RADIATION

CONDITION

Let us assume that the problem we want to solve is the electrostatic potential problem, namely Au=0

(2.1)

in ft

u=g

on r

u-logrbounded Although

we are dealing

asr

with the Laplacian

should keep in mind that the spectral describe is particularly case of operators

_oo. operator,

procedure

the reader

we are going to

designed for those situations

- such as the

with variable coefficients approaching

a constant

state at infinity - where the integral equation method is not feasible. The starting point for deriving several families of radiation

con-

ditions is the expansion of the exact solution into a convergent ries of eigenfunctions, neighborhood

usually through

separation

of variables

sein a

of infinity, where the differential operator has constant

coefficients. For (2.1} we have (2.2)

u(r,p)=Iogr+_

ak e,k_

(r, p) being the polar coordinates

in the plane. Note that the right-

hand side satisfies the radiation

condition

ak are unknown.

at c_.

The farfield conditions are obtained

The coefficients by eliminating

these constants, or a finite number of theme on the artificial boundary too.

Bayliss, Gunzburger ators in this process.

and Turkel use in [BGT] differential

oper-

Let us briefly recall their far-field conditions,

which we also implemented in a spectral context. The idea is to differentiate (2.2) m times in the r direction and eliminate the ak's for ] k I__ m through a linear combination of such derivatives. For each m __1, this yields a differential operator

(2.3)

B,_= _'-1 _+

r

which exactly anihilate the terms of order up to 2m in the series (2.2), i.e.,

(2.4) Bm(.-logr-a0)=0 r,m+l ,m=1,2,... with a0 estimated by averaging u in the farfield.

The approximate

solution uap is required to satisfy the farfield condition

(2.5)

B_(_°_- logr- a0)= o

on r_

By comparing

(2.5) with (2.4), it is seen that this method produces

an "a priori" (i.e., independent of the numerical discretization) error on the approximate solution,

due to having dropped the terms on

the right-hand side of (2.4). This error decays algebraically with the distance of the artificial boundary. It can be made arbitrarily small by raising the order of the farfield operator, but this may lead to a cumbersome or inefficient numerical procedure. The alternative approach which we follow in our construction of an infinitely accurate boundary operator consists of expressing each coefficient ak as a functional of u, rather than eliminating a finite number of them via a differential operator.

Observe that for any

r > 0, ak/rlkl

is the k-th Fourier coefficient of the periodic function

_-+u(r, _o), as (2.2) shows. Thus the following representation

(2.6)

rlkl ak -- 27r 1 f2_ u(r,O)e-ik°dO

If we differentiate

holds

-- ilk(r).

(2.2) with respect to r:

r

[

o.]

k

and use (2.6), we _et an integro-differential

relation satisfied by the

exact solution of (2.1) on every circle of radius r. Precisely we have

'[,'/0

u,(r, lp) =r

2-_

]

}_-" l k l eikC_'-°)u(r'O)dO k

or

1 Ur =---

(2.7)

1

r

r

K,u,

where • denotes the convolution of u with the singular kernel OO

K(y)We impose the radiation

_1 Ek

condition

cos kr/"

k--1

(2.7) on the approximate

solution

over the artificial boundary 1"_. Clearly, the precision of this farfield condition,

as well as the efficiency of its implementation,

rely upon

the accuracy and the easiness with which the singular integral in (2.7) is evaluated. boundary approximate

Accuracy

and efficiency are guaranteed

if the artificial

is a circle, which is not at all a restriction, solution is a trigonometric

polynomial

and if the

on too, which is

the case if a spectral Fourier method is used in the angular direction.

Assume thatroo = {(r,_o) I,Ir I= !!co)and thatthe approximatesolution, which we denote by u N is represented

on rco by a

trigonometric polynomial of degree N

_:_ (Rco, _)= _ _(R_)_''_ Ikl_