I nternat. J. Math. Math. Sci.
No. 2 (1980) 11-50
311
Vol.
A FINITE ELEMENT METHOD FOR EXTERIOR INTERFACE PROBLEMS
R.C. MACCAMY
Department of Mathematics
Carnegie-Mellon University
Pittsburgh, Pennsylvania 15213
S.P. MARIN
Department of Mathematics
General Motors Research Laboratories
Warren, Michigan 48090
(Received January i0, 1979)
ABSTRACT.
A procedure is given for the approximate solution of
a class of two-dimensional diffraction problems.
Here the usual
inner boundary conditions are replaced by an inner region to-
gether with interface conditions.
The interface problem is
treated by a variational procedure into which the infinite
region
behavior is incorporated by the use of a non-local boundary
condition over an auxiliary curve.
The variational problem is
formulated and existence of a solution established.
Then a
corresponding approximate variational problem is given and
R. C. MACCAMY AND S. P. MARIN
312
optimal convergence results established.
Numerical results
are presented which confirm the convergence rates.
KEY WORDS AND PHRASES. Approximation Property, Approximate
Variational Problem, Convergence, Convergence Rate, Elliptic,
Finite Elements, Galerkin, Helmholtz Equation, Integral
Equation, Optimality, Potential Theory, Regularity
65M05, 65MI0
1980 MATHEMATICAL SUBJECT CLASSIFICATION CODES.
i
INTRODUCTION
In [9] a method was presented for the numerical solution
of some diffraction problems.
We believe this method, a
combination of variational procedures and integral equations,
to be of quite wide applicability.
To illustrate the method
we discuss here an exterior interface problem for the Helmholtz
The main idea is to use integral equations to reduce
equation.
diffraction problems in infinite regions to variational problems
over finite domains but with non-local boundary conditions.
In
section four we indicate how the general method can be adapted
to other situations.
Let
be a simple smooth closed curve dividing
into two open sets, a bounded region
region
for some
n 2.
R
(i.e., we assume that
>
+This work was
0.)
n2
QI’
D
and an exterior
{x 6
]R
2
ix >
R]
We begin with the problem
supported in part by the National Science
Foundation under Grant MCS 77-01449 and in part by the Office
of Naval Research under Grant N00014-76-C-0369.
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
u
Find
such that
AU +
k2]u
f
in
Au +
ku
0
in
u(x
(Xo)
lim r
rco
Here
I’2 >
0
,
the
u (x
0.
ik2u
are constants,
the direction of
1
2
o)+, XoF
u
+
Xor
2 (Xo)
U(Zo )-
()
313
n
denotes differentiation in
unit outward normal to
O)
F and, for
u (x)
lim
x_Xo
(xO)
U(o
+
lim
xx o
u(x)
We attach similar meaning to the notation
numbers
k 21
and
k
2
2
(Xo).
are complex constants with
k
Also the
2
satisfying:
Im(k 2)
0
and
Re(k 2)
>
0
Figure i. 1
if
Im(k 2)
O.
R. C. MACCAMY AND S. P. MARIN
314
We refer the reader to [2] and [8]
for existence and uniqueness
results for problem P.
A physical realization of problem P is found in electro-
magnetic theory when the scattering of a time periodic incident
wave from an infinite cylindrical conductor is considered.
(See for example [4] .)
The method that we present here is a mixed variational-
integral equation technique for the interface problem.
It is
based on the introduction of a boundary condition which enables
problem P to be reduced to a boundary value problem over a
2
bounded subregion of ]R
The boundary condition is described
as follows.
F
Let
exterior by
be a simple smooth closed curve and denote its
Aco.
It is a classical result that the problem
AU +
k2u
(Qk)
on
u
lim r /
roo
u
I
has a unique solution for any
F
problem
C.
(Qk)
For given
by
,
ikul
k
k, and
Uv (x;k,)
A
in
0
0
C, k
6
Foo
for
x
0
and for any function
we denote the solution of
6
Aoo
and define the
for
Xo6
operator
Tk[] (x) =-
BuF
I
"5n (Xo;k’)
+
o"
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
We observe that the solution
u
315
of problem P is also a solution
of the boundary value problem given below provided the curve
is chosen so that its interior contains
()
u +
k]2u
f
in
QI
Au +
ku
0
in
Q2
u
-
(xo)
Bu
To
T
Q2
Here
Q1
(x o)
Foo
U F.
T
o) +
u
+
2 (Xo)
u (x
[u
Io
(_xo),
denotes the annulus that lies between
F
and
oor"
This is shown in Figure 1.2.
Figure 1.2
The validity of the boundary condition
for
x e F__
o
Bu
u
and
Conversely, if
u
is a solution of problem
n
are continuous across
to a solution of problem P by defining
6
A
.
(Xo)
for the solution of problem P follows from the
fact that both
x
Tk[UIF
(Xo)
u
we may extend it
U (x;k2,ulF
for
R. C. MACCAMY AND S. P. MARIN
316
From here our plan is to approximate the solution of the
The principle
problem P using the finite element method.
ingredient in the finite element method is a variational
formulation of the problem which, here, we construct in a
straightforward manner using Galerkin techniques.
v
C
6
u
Then for
(QI U Q), vlQ 1
6
C
is a trial function with
v
this development by assuming that
We begin
I(Q),
v
c (n).
nT2
we have
the solution of
Integrating by parts and using the interface conditions together
with the boundary condition on
2
rooTk2
Woo
we conclude that
oo
i
J QI fv
We denote the left hand side of 1.2 by
hand side by
F(v).
Thus,
u
dx
for all such
a(u,v)
and the right
the solution of problem P
satisfies the variational equation
v.
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
U
T
2 ),
v
(1.3)
F(v)
a(u,v)
i
(i),
v
With a variational formulation of
P
v
for all
6
(i
C
6
317
C
C
6
over a bounded region
available, the next step is to introduce approximate variational
To do this we select a finite number of trial functions
h
h
h
h
span[l,...,N ]" We then attempt
I’2’’’’’N and set S
h
h
to find a function u 6 S
which satisfies
problems.
h h
a(uh vh)
The solution
u
h
F(vh)
for all
v
h
e S
h
(i .4)
of (1.4) is taken as an approximation to
u.
The complication in the above procedure is the determination
of the operator
Tk.
This can be done by integral equations and
in particular by using integral representations for the solution.
U
F
(x;k,)
of problem
U (x;k, )
obtain
in the form
U (x ;k, )
where
O
k (x,y)
It is shown in [6] that one can
Qk"
i/4 H O(i)
j
(1.5)
y)ds y
(y) Gk (x
N’
(klx-yl),
(I)
Ho
function of the first kind of order zero and
is the Hankel
is determined
by the equation
Kk[] (x)
r
(y)Gk(X,y)dSy
(,), ,DeFoo.
(1.6)
R. C. MACCAMY AND S. P. MARIN
318
Equation (1.6) is a Fredholm integral equation of the first kind.
It is shown in [6] that this .equation is uniquely solvable and
Kkl[
we set
From the representation (1.5) and a
standard result in potential theory one has
+
5UF
n
+
(Xo)
(Xo;k’)
Gk(Xo;Y)ds
(Y)
(1.7)
1
(
I +
Mk)[] (x O)
for
Thus we have the following characterization of
k[]
(]
+
k
oX 6oo.
T
k
[].
The remainder of the paper proceeds as follows.
In
section two we describe the variational procedure precisely,
and we state the convergence results.
reduced to two coercivity inequalities.
The proof of these is
The verification of
these is extremely technical and postponed to section five and
the appendix.
In section three we discuss the implementation of the
method including an approximate treatment of the operator
Tk.
We report on some numerical experiments which confirm our
estimates for convergence rates.
Section four contains a brief discussion of other problems
to which the method applies.
The authors wish to express their appreciation to
Professor G. J. Fix for his help in the development of this paper.
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
2.
VARIATIONAL F ORMUI2%T ION
For any region
Z
of order
i
k
II"k
For closed curves
3
the space of
with square integrable derivatives
Z
and we write
2
llfllk’Z
6
Hk(z)
we denote by
complex valued functions on
r
319
7
Z
2dx
IDf
that if
there are continuous mappings
-
{2.1)
r
H (7),
we also need the boundary spaces
It is known (see [i]
u
f6Hk (Z)
for
-
k(z)
ulB Z
H
BZ
is smooth then
k-I/2(Sz).
(2.2)
H
For our variational formulation we will need a space
which
we define as follows
[vIvIozH z(o z),vlo2T
We also define the norms
IIIv III 2j
6H
1
v+ (,X..O)
T
(n2) ,v (,.)
IIl’lllj
2
Ilvllj,
on
,.6"["]
by
3,Q T2
is a Hilbert space under the norm
and note that
II1"
IIz.
To proceed with the variational formulation we need the
following properties of
LEMMA i.
to
H
r- 1
(Fco)
T
k
Tk,
which are proved in the appendix.
is a bounded linear operator from
and satisfies
Hr
(Foo)
320
R. C. MACCAMY AND S. P. MARIN
r
,@
for all
6
Tk()@
r
ds
(2.3)
Tk($)ds
I/2(FO)
H
With this we observe that the bilinear form
a(u v)
2 F Tk 2 (uIFoo )v
co
ds
i
n
k2
lUV)-- d
(Vu.v
1
(2.4)
.
is well defined on
El/2 (oo)
By Lemma i then
By {2.2)
IF
T (u
k
(u-v
k2uv) dx
n2
u
IF
and
are in
H-I/2 (oo)
is in
’oo
co
2
2
with
oo
Hence by the generalized Schwarz inequality and (2.2)
I[ F
Thus
2
(ul)
OO
a(u,v)
cllull/2
sl <
rOO llvll/2 rOO <
c,
lllullllllvlll.
is well defined and
(2.6)
showing that
a(.,.)
comment here that if
HE
f
6
L2
is bounded.
C
(l)
H
F (v)
fv
1
is a bounded linear functional on
(l)
.
d
then
We may also
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
The variational form of problem
321
that we use is stated
as follows
u
Find
(V)
such that
6
a(u,v)
F(v)
for all
v
HE
6
Next we state the approximate problems.
[SEh]0<h<l
H
E
We suppose that
is a family of finite dimensional subspaces of
which satisfy the following:
APPROXIMATION PROPERTY.
and positive constants
with
lllulll <
C
o
i t,
D,
There exists an integer
and
C
1
t
such that for any
there exists a function
u
2
u
6
6
S
H
E
h
E
which satisfies
III u-ullIj
(the constants
C o, C
With such a family
1
(AVe)
j
(2.7)
0,i
h
are independent of
IS hi
E
Find
Cj ht-J III ulll 6
i
and
u)
we pose the approximate problems:
h
u
a(u
e S
h,vh)
for all
h
E
such that
F(vh)
h
v e S
h
E
Our main results concerning problems
V
and
AV
are:
R. C. MACCAMY AND S. P. MARIN
322
THEOREM I.
V
There exists a unique solution
and there exists an
h
u
unique solution
<
h
such that problem
0
whenever
h
h
of problem
AV
has a
< h o.
There exists constants
THEOREM 2.
that, for
>
h
u
C
and
o
C
1
>
such
0
o
III u-uhllll
CllllU-Whllll
h
for all
(2.8)
Wh6S E
and
h
u
The constants
C
u
and
o
C
1
h
(2.9)
are independent of
u
h
and
<
h
o
Theorem 2 is an optimality result and is typical for finite
element methods applied to elliptic problems.
approximation property of
u,
of the solution
COROLLARY i.
then for
of
<
h
h
h
o
<
h
IS hi,
E
together with the regularity
we obtain
Suppose that
f
6
H
(n2T
H
-2
(n I)
with
2
j
0, i,
independent
Cj,
i
i t
such that
PROOF OF COROLLARY i.
uln
6
there are constants
Cj h’-j
III u-uhlIIj
show that if
If we use the
H
f
2
(i)
.
Regularity results for problem P
then
thus
Ill ulIl <
inf
h
E
IIlu-whllll
Wh6S
(2 .i0)
(D
u
IQ 1
6
H
(QI)
and
By (2.7) we have that
C1 hg-1 III ulll,j.
323
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
Applying this in (2.8) we find that
lllu-uh III <
ch-I lllu llI.
(2 .ii)
From (2.9) and (2.11) we obtain (2.10) for j
III u-u IIio <
0, i.e.,
Co lllu III.
The proofs of Theorems 1 and 2 are complicated by the fact
that the variational problem is not positive definite.
Hence
we need the following rather technical result which is treated
in section five.
THEOREM 3.
sup
0v6 HE
sup h
0v s.
ho,C a >
There exists constants
la(u,v)
Illvllll
IIlv Ill
>
C
>_
Ca
a
0
Illulll 1
for all
u6H
Illuh Ill m
for all
UheS E.
such that
(2 12)
E
h
(2.3)
Once (2.12) and (2.13) are established we may use these
estimates in a standard way to prove the existence and uniqueness
Before proving (2.12)
results stated in Theorem 1 (see [I]).
and (2.13) we will first show that 2.13 yields 2.8 of Theorem 2.
(The result 2.9 will be treated in section five.)
From the formulation of problems
a(u,vh)
Wh
e S
h
E
F(vh)
a(uh,vh
for all
V
v
h
6
and
h
S E.
AV
we have
Thus, for any
R. C. MACCAMY AND S. P. MARIN
324
la(uh-wh,vh)
(2.14)
The right hand side of (2.14) is bounded above (using (2.6)) by
C
lllU-Whlll I.
By taking the supremum over
v
h
6
h
SE,
v
h
0
and applying (2.13) we obtain
llluh-wh IIII i
C
’lllu-wh III i"
lllu _uh IIIi
The triangle inequality applied to
III u-uhllll
i
llIU-Whllll
+
(2.15)
gives
IIluh-whllll"
Using this and (2.15) we obtain
Ill u-uhllll
i (I + C’)
lllu-whllll.
Thus (2.13) implies (2.8).
3
IMPLEMENTATION OF THE METHOD
The approximate problem
F ind
uh
6
S
h
E
such that
a(uh,vh)
(AVe)
for all
F(vh)
h
VheS E
is seen to be equivalent to a matrix problem by selecting a
h
for S
basis
{I’2’’’’’NH]
E.
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
uh
We find a function
325
given by
u
q_.
Z
_.
(3 .i)
j=l
which satisfies
a(uh,i)
(,i),
i
(3.2)
i,...,h.
The system of equations (3.2) is the matrix problem
(3.3)
where
(ql,...,qNH)T
q
f
is the vector of weights in (3.1),
H))T
(F(I)’’’’’F(N
is the source term
and
K
(Kij)
is the stiffness matrix
with entries
Kij
a(j’i)
T(jlFoo)ids
2
oo
(V’j’i
i
k21ji) dx
1
(3.4)
To use the ideas presented so far in actual computation
we must be able to impose the nonlocal boundary condition
u
(u
I)
R. C. MACCAMY AND S. P. MARIN
326
oo"
along the outer boundary
We see from 3.4 that, in the
approximate variational problem, this amounts to computing the
integrals
[
F
Tk(ilroo )ids
for the basis functions
"’Nh
1’02’
(3.5)
of the approximation
h
This computation may be carried out in a straightE
forward manner according to the definition of T
k by solving
the integral equations
space
S
J
r’
i (Y)Gk(X’y)dSy
for the densities
i’
xeFoo
i (x),
computing
i
(x)
Tk (i)
I,...,Nh
for
x6Foo
(3.6)
from
the formu i a
Tk[i (x)
@i(x)
+
i(Y)
Gk(X’y)dSy
(3.7)
and finally computing the integrals 3.5 using a suitable
quadrature rule.
The execution of this procedure for general
finite element spaces
h
SE
is a lengthy process at best
Fortunately the matter of computing the integrals 3.5 can be
greatly simplified by making special choices of the curve
h
and the approximation spaces S
In the following discussion
E
we take
to be a circle and choose S h so that the
E
F--
restrictions of the trial functions to
F
are piecewise linear
functions of arclength corresponding to a uniform mesh along
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
_
T
2
Figure 3.1 shows the region
radius
h
S
E
R.
when
Foo
span[l, "’’’Nh
has been chosen so that
vanishes identically along
Foo
is a circle of
We assume that a finite element space
is that
i
Foo
piecewise linear function of arclength along
on
327
which either
or is equal to one at one node
and vanishes at the remaining nodes.
Figure 3.1
If we set
8j
hlJ
characterize each
on
(D)
function
1,2,...,N l
j
i
IF
where
where
2
I
N1
is the
$o
as a translation of a
8
27r-periodic extension of the
function defined by
-e/h
o()
we may
(which does not vanish identically
in terms of the polar angle
$o (8)
h
+
e/h I + 1
0
<h
-h i e < 0
I
h & lel <
0
<
.
R. C. MACCAMY AND S. P. MARIN
328
We have, for those
cosS,R sinS)
i(R
for some integer
which do not vanish on
i’s
we may assume that
m
i.)
i
N I.
mi
i
mi,
$o(8
Woo
(3.8)
hlmi)
(By renumbering the
i’s
We note that, by solving elementary
boundary value problems for the circle, one obtains the formulas,
H()
Tk(COS(n8
for
n
0,i,2,..
V (kR)
k _.n
cos(n8
(1)
Hn (kR)
+ ))
where
n.
first kind of order
Hn(I)
+ )
is the Hankel function of the
(The superscript
"V"
differentiation with respect to the argument.)
expands
@o
denotes
Then, if one
in a Fourier series one obtains, after some algebraic
rearrangement, see [9],
(3.9)
2v
F
h21 + h [Tk[P] (8))I e= (-2) h
where
(8)
f(8 + h I)
f(8)
is the forward difference operator,
and
p(8)
"n"
4
9--
"n"
2
I,,el 2 + lel 3
12
12
le[ 4
48
181
.
21r.
(3.10)
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
329
From the formula 3.9 we see that to compute all of the
integrals 3.5 in the special case under consideration we need
only compute
function
Tk[p] (8)
p(8)
at
%j,
I,...,N 1
j
for the single
This amounts to first solving
defined by 3.10.
the integral equation
2
R_i4 (t)H(1)o (2kRlsin ---tl)dt
8
P(8)
0
2
8
(3.11)
0
for the density
F
(t).
(We have specialized to the case when
is a circle of radius
i
Gk(x,y)
x
when
on
Foo.)
R
H(1)(2kRlsin
8___tl)
o
(R cosS,R sinS),
Once
(t)
(R cos t,R sin t)
i t i
0
is found from the formula:
Tk(p)
(8)
1
-
k
(8) + Ri
and used the fact that
27r
is determined
are points
T k(p) (8)
(again specialized to the case when
(t)Hl(1) (2kRlsin 8---tl)Isin 8--tldt.
0
The kernel of the integral operator in 3.8 is obtained using
the fact that
%nx Gk(X,y)_
when
x
-ik Ho(1)V(2kRlsin 8__tl) isin 8__t
(R cosS,R sinS)
and
y
(R cos t,R sin t)
together with the identity
o
(.)
_H()(.).
are on
(3.12)
R. C. MACCAMY AND S. P. MARIN
330
We may also observe that this kernel is continuous at
t,
8
in fact,
lim
8t
HI) {2kR
8---t I)
sin
sin
_i
8---t
7rR
In the numerical examples that follow the equation, 3.11
was solved using numerical methods described in [6] with
Simpson’s rule replaced by the rectangular quadrature rule.
With this modification the discretized form of 3.11 is a matrix
problem
p
A
with
A
a circulant matrix.
This feature enables the problem
to be solved efficiently using well known inversion formulas for
circulants (see [5] or [I0]).
To verify the convergence rates predicted by the theory we
consider the following example for various values of
k
I
k
i’ 2’
2
Find
u
such that
au +
klu
0
nu +
ku
0
U
1
+
U-
U
,u, +
=2t’
lim
r-D
< Ix <
l_xl >
on
on
l_xl
l_xl
2
2
I
(3.14)
2
u
on l_xl
I()Bu
rl/2 Iik2ul 0
2
331
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
F
In this example the curve
greater than two (we use R
is a circle with radius
Following our procedures we
3).
construct the boundary value problem
Au
+
k]2u
0
Au
+
k22u
0
u
1
on
u+
u
on
n
1
2
i ()
on
I_xl
=
Tk 2 [u I--xI=R
on
Ixl
3.
2 ()
B__u
lxl
Ixl
+
Figure 3.2
The approximation spaces
IS h]
E
used here are sets of piecewise
linear functions of the polar coordinates
[xll <
constructed by first mapping the region
the rectangle
[0,2] x [I,R]
in the
r,8.
r-8
They may be
Ixl <
R]
into
coordinate system.
We then construct piecewise linear finite element spaces (composed
of 2-periodic functions) corresponding to triangulations of
the rectangle and transform back to rectangular coordinates.
.
We
thus obtain a distorted triangular grid with associated trial
functions which are linear in
of subspaces
IS hi
E
(n
r
and
The resulting family
maximum diameter of the triangles)
satisfy the approximation property with
Corollary 1 we should observe that
t
2.
According to
R.C. MACCAMY AND S. P. MARIN
332
-u
h
0
(h
2)
and
_u
IShi
E
for this family
h
Our examples are chosen so that the
exact solutions are known and we measure convergence rates by
I is the
I
where u
and
computing
i
III uh-uI III
llluh-u IIio
interpolant of the exact solution in
S
h
E
From approximation
theory we have that
iil _u lifo
0
(h 2)
III u-uI III 1
and
0
(h)
Using this and the triangle inequality we may show that the
h
-uh 1 will be optimal order
and
errors lllu-u
(0(h 2)
and
III
lllu
IIio
0(h),
respectively) if we observe in the
calculations that
uI
IIio
c h
o
2
lluh-uIllll < Clh
and
(3.15)
We display the results graphically in Figure 3.4 and Figure 3.5
by plotting
III uh-uIlllj
on a log-log scale.
vs
A slope of
(linear) convergence.
i/h
j
-2 (-I)
0,I
indicates quadratic
Eight trials were conducted.
The values
used to solve problem 3.14 in these
2
cases are listed in Table 3.1.
of
i’ a2’
kl
and
k
333
FINITE ELEMENT MEHTOD FOR EXTERIOR PROBLEMS
TRIAL
61
2
kl
k2
i
i
4
i
2
2
i
2
I
2
3
i
i
4
4
i
4
2
i
4
5
i
4
i
i0
6
2
4
i
i
i
i
4
4
4
i
i
i0
7
8
Table 3.1
Figure 3.4 shows
h-uIlllO
lllu
every case, for sufficiently small
quadratic.
In Figure 3.5
I/h
vs
h,
llluh-ulllll
and we observe in
that the convergence is
is plotted against
Here the slopes of the curves lie between
-I
and
-2
i/h.
indicating
that
llluh -u I III
i
h.
Thus, from the remarks preceding 3.15, we observe that the
convergence rates are optimal.
4.
EXTENS IONS OF THE METHOD.
The particular problem studied here was chosen for
illustrative purposes only.
It demonstrates the power of
variational methods to handle complicated situations on finite
regions and the ability of integral equations to deal with
infinite regions.
We sketch a few more examples.
R.C. MACCAMY AND S. P. MARIN
334
mO
0
0
0
0
x
HOIdH3 H
--o
o
o
b
b
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
335
We note first that all the standard exterior problems for
the Helmholtz equation can be treated.
k2u
Au +
the problem
in
0
2
That is one can solve
with the radiation condition
and any combination of Dirichlet, Neumann or mixed data given
on
F2.
The next observation is that the exterior problem can also
be treated in the case of variable coefficients.
Consider the
equation,
div(A(x)Vu) + b(x)N
Suppose there is an
A
1 )’
k2(x)u
(0’0’0)’
bN
F
+
such that for
O
i 0
(0
Then if one chooses
Ixl
R
"Vu
k2
in
0
lxl >
RO
(x)N
k 22.
2"
(4.1)
we have
(4.2)
so that it contains the circle
one can formulate boundary value problems for 4.1,
R
with the radiation condition as variational problems over
with the condition
U
n-
Tk
2
(uIFoo
on
T
2
too.
An example of equation 4.1 occurs in [3] in the study of
acoustic radiation from a cylinder when heating causes local
spatial inhomogenities.
The equation there is
c 2 5t 2
c(x)
Zp + 1 V)’Vp
where
c
p(x,t)
is the acoustic pressure.
is sound speed,
solutions of the form
at 4.1 with
P(2t)
D
(x
0
(4.3)
is the density and
If one seeks periodic
Re (u(x))e it then one arrives
R. C.
336
A=
K4_CCAI
i 0
(0
1 )’
AND S. P. MAN
,
I
b=N
0
c
Finally it can be seen that one can treat interface
problems with the geometry of Figure i.i, but with equations
of the form 4.2 holding in
(Al,b_l,k I)
hold for
and
(A 2
for
with associated
It is necessary only that 4.2
(A2,b2,k2).
b2 k 2)
2
and
i
II
R
and that the second
O
interface condition be replaced by one which is naturally
associated with 4.1, that is,
2(A2Vu" nA) +"
l(AlVU’nA)5.
PROOF OF THEOREM 3.
We begin by considering an auxiliary problem.
Find
AU
u
such that
0
u=
Qo
u
0
in
A
on
V
co
()
r co
as
Vu
This is:
0(r -2
This problem has a unique solution which we denote by
and we may define the associated
Qk"
T
operator
That is
To [’] o
o ;)
’n
To
o
U
(x;)
as in problem
337
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
The following results concerning the operator
To
are
established in the Appendix.
Hr-i (co)
is a bounded linear operator from
TO
LEMMA 2.
H r (1oo)
with the following properties.
(i)
I To(,)ds
(ii)
To()ds
For all
(iii)
from
Hr
To(@)ds
for all
0
k,
Tk
(FD)
for all
To
into
,,@
e
is a bounded linear operator
Hr+l (co)
We
With Lemma 2 stated we can outline the proof of Theorem 3.
write
to
a (u, v)
in the form
a l(u,v)
a(u,v)
(5.2)
+ a 2(u,v)
where
oo
V
a
2(u,v)
02
Ol
I 2(T-To) [uIF
and look for a
v e
2
O1
of the form
which the inequality 2.12 holds.
the decomposition 5.2 and find
V
I udx
(5 4)
OT
2
u + w, w
For this function
for
v
we use
R. C. MACCAMY AND S. P. MARIN
338
T2
1
+ [a 2(u,u) +
a
ud +
I
a
1
2Judx]
(5.5)
+ a(u,w)
The first bracketed expression on the right side of 5.5 is
negative and bounded above by -C’ lllu 2
This follows from
i"
III
Lemma 2(ii) and the definition of
a
I(.,.)
If
w
can be
chosen so that
-[a 2(u,u) +
a(u,w)
then we would have for
v
u
i
w
udx +
a2
n;T2
C’
=
Illu III 1"
llllllx _<
(x +
(5.7)
satisfies the estimate
IIIwlllx c,lllulllx
then
(5.6)
+ w,
a (u v)
If, in addition,
nI
c,)Illulll
(5.8)
and it would follow from 5.7
that
l.a (u, v) > x
lllvlllx
proving 2.12.
c
/
c
Illulllx
Inequality 2.13 follows from 2.12 and the
approximation property for
IS hi
E
if the estimate 5.8 can be
strengthened to
lllwlll= < c’lllulllx,
(5.9)
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
To see this we set
u
u e Sh
h
E
and construct
339
so that
w e
Then
5.6 and 5.7 hold.
(5 .0)
holds for
uh + w.
v
By the approximation property 2.7 and
the assumption 5.9 we may pick
If we set
vh
uh + wh
h
wh e S
E
such that
then there exists
la(uh,)l >_ c" Illuh III
2z
h
Moreover, using 5.9 and 5.11,
IIllllz _<
Illuhlllz
+
Illuhlll
+
-
>
< ho.
h
for
O
(5 .)
0
such that
(5.12)
IIllllz
IIIwll12 + IIIw-lllx
_< ( + c, + cxh) Illuhlllx.
Thus
IIIv III
_< c’" Illuh III
.
(5.13)
Finally, we note that
h)
a (Uh,V
c Illulllz
IIIvhlllz >-
follows from 5.12 and 5.13.
for
h
<
h
o
This proves 2.13.
This
result is a simple consequence of 5.10, 5.11 and 2.6
using the estimate
la(uh,Vh) >_ la(uh,V)
la(h,W-Wh) I.
340
R. C. MACCAMY AND S. P. MARIN
TO complete the proof of Theorem 3 we must show that for
arbitrary
hold.
u
there exists
E
w
such that 5.6 and 5.9
E
To do this we consider the variational problem
w e
Find
-[a 2(O,u) +
a(e,w)
such that
’i
I
for all
e
.
I )dx
n
+
2
,.
dx]
Our arguments to this point utilize what is known as Nitsche’s
Trick [11], and the problem VP
that arises in these instances.
.
is the sort of adjoint problem
The desired results 5.6 and
5.9 are immediate consequences of an existence and regularity
result for VP
This is stated in the following 1emma which
is discussed in the Appendix.
LEMMA 3.
Moreover,
w e HE satisfying problem VP
and satisfies
There exists
III wll12 <
c
IIIwlll 2 _< c,lllulll.
Having completed the outline of the proof of Theorem 3
2 estimate 2.9 of
we return to the matter of proving the L
u
uh
Theorem 2. We again use Nitsche’s Trick. Let eh
and consider the problem
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
Find
w e
341
such that
(5.14)
v e HE.
for all
We may show, using integration by parts and Lemma i, that this
is equivalent to the following boundary value problem
kl2W
+ k22w
+
w
C
1
CZ
Q1
in
Q2
2
+
()
Tk 2 1 Foo )-)-
T
r
on
w
()
in
r
on
o
F
on
This may in turn be recast as an exterior interface problem
+
kl2W
w
Cl ()
lira r I/2
--+
w
in
e
h
in
0
in
on
2 (-)
I
+
Q1
n T2
A
r
on
ik2l
(5 .t5)
r
o
R. C. MACCAMY AND S. P. MARIN
342
Problem 5.15 has a unique solution and, by arguments similar
to those outlined in the discussion of the proof of Lemma 3,
its solution satisfies the estimate
IIIwlll 2
We put
v
e
h
c’lllhlllo.
in 2.14 and obtain
II1%111o.2
a(eh,w)
Since
we
e
h
h
SE
uh
u
we have
(5.16)
a(uh,wh)
a(u,wh)
and
a(e,w)_n
n-
0
(5 17
for all
F(wh)
w e Sh
h
E.
for all
We subtract
this form 5.17 to obtain
(%,w-w)
Ill%lifo2
for al i
Whe
h
SE
(5 .8)
From 2.6 and 5.18 we have
IIlh IIio2
c IIlh 111
IIIw-w] III z
for all
w
h
S
h
E
or
lllh IIIo2
c Illh II!
IIw-w]lllz.
inf
WheS
h
E
(5.19)
The approximation property 2.7 implies that
inf
h
E
IIIw-wh III x I
cx Illw III 2"
WheS
Using this and 5.16 gives
inf
h
E
WheS
IIIw-w] Ill
c
, Ille
h
IIio,
(5.20)
343
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
Finally, 2.19 and 2.20 establish
L2
which prov4s the
APPENDIX:
estimate 2.9.
PROOFS OF. LEMMAS.
PROOF OF LEMMA I.
linear map from H r
(lco)
F
When
inverse.
n Gk
quantity
It is shown in [7] that Kk is a bounded
onto H r- 1 (co) with a bounded
is a smooth curve it is known that the
in the definition of
1
is a smooth function
and the first statement of Lemma 1 follows.
establish, the property
In order to
Green’s theorem argument.
and
V
U
by
U
F
,
Suppose
(k,),
V
UF
2.3 for
e H I/2
Tk
we use a
(Fco)
(_x;k,).
Then
Define U
Green’s
theorem yields
FR
where
is a large circle (radius
R) containing
Fco"
The radiation condition implies that the limit of the right-hand
side-as
tends to infinity is zero.
R
Hence the left-hand
side is zero and this is the result stated.
PROOF OF LEMMA 2.
operator
problem
T.
Qo"
We first obtain a representation for the
To do this we need to discuss the solution of
It is shown in [6] that the solution can be
obtained in the form
ur
oo
(A.2)
,)
F
R. C. MACCAMY AND S. P. MARIN
344
is determined by the equations
where
(y) inlx-ylds
Ko[
a (y)
The equation
Ko[]
ds
Z
(A .4)
O.
can be solved for any
X
(A.3)
+ C
condition A.4 determines the constant
C
M.
The
and serves to make
o
UF
bounded at infinity. It is shown in [7] that KO is a
with a bounded
bounded map from H r (Foo) onto
Hr+l(oo)
inverse.
In order to establish the results in Lemma 2 we must
look a little more closely at the solution procedure (A.2)-(A.4).
From A.3 we have
= Ko-I[]
and then A.4 determines
C
(-
C
+
CKo I[I]
(A.5)
by the formula
[ K l[]ds)/( K l[l]ds).
(It is shown in [6] that if
Fco
(A.6)
is chosen so that its mapping
radius is not one then the denominator in A.6 does not vanish.)
1
-i
we observe that A.6
into
maps
Since K
r
defines C as a continuous linear functional on H
Hr (oo)
Hr- (Foo)
(Too).
Indeed by the generalized Schwarz inequality we have
Icl
Ic(m)
(r-l)
,FO0
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
In order to determine
To
we observe that by A.2 and A.5
’an (--Xo;)]
TOIl] (;&o)
345
u(-Xo)
+
inl--x-xlds Z
)
(A.8)
(I + MO)K l[q)] O
+ C()
(I + MO) K l[1] X),
It follows from this formula that
r- 1
H
continuously.
maps
TO
H
r
FOO
e
X
O
(FcD)
into
(oo)
Property (i) of Lemma 2 follows by the same Green’s theorem
type result as in Lemma i.
The negativity result (ii) is
o
Ur,
another Green’s theorem argument. We have (for u
T O []
FR
Here
ds
Vu
u ds
nR
is as before and
Aco
N
=dx
+
int(FR)
(x ;) ),
(A.9)
u.
Once again the
conditions at infinity imply that tne limit of the integral over
FR
as
R
tends to infinity is zero hence we obtain (ii).
It remains to establisy (iii) of Lemma 2.
observing that
singularity.
Gk(X,y
Rk(l-,.v[)
in
lx-f
begin
by
have the same
We have in fact,
G (x,y)
k
with
and
1
We
.
+
i
2v
inlx-yl + Rk
Ix-ml
[,,,x-,yl21nlx-x.ylyk(I,,X.,X-,.,y[)
(A. X0)
+
6k(I,,,x-.yl)
(A.11)
R. C. MACCAMY AND S. P. MARIN
346
where
is a constant and
k
and
7k
8
k
are analytic.
Thus we may write 1.6 in the form
(A.12)
(y) In
or,
It is shown in [7], on the basis of A.II, that the integral
operator
c’)Rk (I,,.x-yl)as M
takes
Hr (co)
Hr+3 (Fco)
into
Hence if we compare A. 13 with
A.5 we see that
U
o
U
(_x;k,)
(x;) + .C()
2v
I KI
[i] in
Ix_-_y IdsX
(A. 14)
+
,I Gk(’) c(Z)Rk(l-l)dszds"
UF
Now we obtain
operator
that
Tk
by computing
.n(D)+
This introduces the
Mk as in 1.7. we note, however, that A.12 implies
differs from MO by terms with more regularity
Mk
(specifically
Mk-M
Hr (D)
Hr+2 (oo)
If one performs
the calculations with A.13 and A.8 one finds that
Tk[]
T o[]
C()
(I + Mo)Kol[l] + k []
FINITE ELEMENT METHOD FOR EXTERIOR PROBLEMS
k
where
is a continuous map from
Hr
(oo)
into
347
H
r+l
Now, by A.6 the functional is given by
F
where
8
is a constant.
But
F
(The constant
FeD
K
-i
is self-adjoint hence,
F
also involves
is smooth then
-i
KO [i]
K-Ill]
o
.)
Now if the curve
would be a smooth function and
then A.16 and the generalized Schwarz inequality yields
IIC()
(I + o)Kol[l]llr+l,Fo
i
IC()lll(I + o)Kol[l]llr+l,Fo
Thus A.15 yields (iii) of Lemma 2.
PROOF OF LEMMA 3.
a (., .)
The definitions of
a(., .)
and
and integration by parts used in a standard way yield
2
that the problem VP is equivalent to the boundary value problem
-
R. C. MACCAHY AND S. P. MARIN
348
kl2W
+ k22w
+
w
w
=2 ()
Tk2 (witoo
+
()
in
QI
in
n T2
(A.17)
on
i ()
r
on
=-(Tk2-TO) (u)
--h
on
Foo.
The result (iii) of Lemma 2 together with the fact that
6
H I/2
(oO)
gives the information that
(A. 18)
-
We may further note that problem A.17 is equivalent to the
exterior interface problem
+
/
w
kl2W
k22w
w
a’2 (’)
w
QI
-u
in
2
0
in
A
on
w
on
h
T
F
(A. 19)
on
i (’)
()
lim r
in
Foo
on
1/21 ik2l
0.
FINITE EL HETHOD FOR EXNERIOR PROBLEHS
349
The solution of A.19 can be obtained in the following form:
(x,z) dSy +
w (x)
in
w(x)
(A. 20)
2 (Y)Gk2 (x’y)dsNZ
+
wl (x)N +
w
2
(x)
in
n2
where
O(x)
w
J" (y)Gkl(X,y)dy
Q1
(A.21)
(A.22)
w 2(x)
I
h(y)Gk
(x,y)dsy.
(A.23)
Standard potential theory arguments show that A.20
satisfies all the conditions of A.19 except the interface
conditions on
F.
The imposition of these leads to the
integral equations,
Kk2[2]
e2
(I Mk2)[G2]
+
+
-
+ wI + w2
Wl + --]
w2
i[i]
1
el[ (-I
+ w
on
F
(A.24)
Mkl)[GI
+
’--]
Here the integral operators are as in 1.6 and 1.7 but on
instead of
(A. 25
w
+
on
r.
F
It can be shown that the solution of A.19 is unique and
the Fredholm alternative can be used to establish the existence
R. C. MACCAMY AND S. P. MARIN
350
of solutions of A.24 and A.25. The estimate in Lemma 3 can be
established by tedious but fairly straightforward analysis of
the mapping properties of the operators in A.20.
We omit
these details.
REFERENCES
i.
Babuska, I. and A. K. Aziz, "Survey Lectures on the
Mathematical Foundations of the Finite Element Method",
The Mathematical Foundations of the Finite Element
Method with Applications to Partial Differential
Equations, A. K. Aziz, Ed., Academic Press, New York, 197 2.
2.
Barrar, R. B. and C. L. Dolph, "On a Three Dimensional
Transmission Problem of Electromagnetic Theory", _Journal
of Rational Mechanics and Analysis 3 (1954), 725-743.
3.
Chernov, L. A., Wave Propaqa. tion in a Random Medium,
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4.
Duffin, R. J. and J. H. McWhirter, "A n Integral Equation
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Franklin
Institu_t__e. 29___8 (5,6) (1974),
385-394.
5.
Greenspan, D. and P. Werner, "A Numerical Method for the
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rch. Ration. Mech. Anal. 2_3 (1966), 288- 316.
6.
Hsiao, G. C. and R. C. MacCamy, "Solution of Boundary Value
Problems by Integral Equations of the First Kind",
SIAM Review 15 (4) (1973)
7.
Hsiao, G. C. and W. L. Wendland, "A Finite Element Method
for Some Integral Equations of the First Kind", Journal
of.LMath..A.nalysis an..d Applications 5_8 (3) (1977).---------
8.
Kittappa, R., "Transition Problems for the Helmholtz Equation",
University of Delaware, Department of Mathematics, AFOSR
Scientific Report, October, 1973.
9.
Marin, S. P., "A Finite Element Method for Problems
Involving the Helmholtz Equation in Two Dimensional
Exterior Regions", Ph.D. Thesis, Carnegie-Mellon
University, Pittsburgh, PA, 1972.
i0.
Smith, R. L., "Periodic Limits of Solutions of Volterra
Equations", Thesis, Carnegie-Mellon University,
Department of Mathematics, 1977.
ii.
Strang, G. and G. J. Fix,
An Analysis. of_ the..Finite Element
Method, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973.