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
,":: :D )
1984
LANGLEYRESEARCH CENTER LIBRARY,NASA I-LA_PIOL% Y!.RGINI_'
3 1176 00518 1863
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1 RN/NASA-CR-172388 OISPLA'l
84N29560**
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ISSUE 19 PAGE 3882 !CASE-84-21 MAS 1.25;172380
UTTL~ HIJTH~ CORP~
fll I ABA:
CATEGORY 64
RPT~~
~JASA-CR-l 84/~i3b/80
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_