On a Boundary Control Approach to Domain Embedding Methods.
In this paper, we propose a domain embedding method associated with an optimal boundary control problem with boundary observations to solve elliptic problems. We prove that the optimal boundary control problem has a unique solution if the controls are taken in a finite dimensional subspace of the space of the boundary conditions on the auxiliary domain.Using a controllability theorem due to J. L. Lions, we prove that the solutions of Dirichlet (or Neumann) problems can be approximated within any prescribed error, however small, by solutions of Dirichlet (or Neumann) problems in the auxiliary domain taking an appropriate subspace for such an optimal control problem. We also prove that the results obtained for the interior problems hold for the exterior problems. Some numerical examples are given for both the interior and the exterior Dirichlet problems.
Introduction
The embedding or fictitious domain methods which were developed specially in the seventies ([5],
[2], [34], [35], [28] or [13]), have been a very active area of research in recent years because of their
appeal and potential for applications in solving problems in complicated domains very e#ciently. In
these methods, complicated domains # where solutions of problems may be sought, are embedded
into larger
domains# with simple enough boundaries so that the solutions in this embedded domains
can be constructed more e#ciently. The use of these embedding methods are a commonplace these
days for solving complicated problems arising in science and engineering. To this end, it is worth
mentioning the domain embedding methods for Stokes equations (Borgers [4]), for fluid dynamics
and electromagnetics (Dinh et. al. [11]) and for the transonic flow calculation (Young et. al. [36]).
In [3], an embedding method is associated with a distributed optimal control problem. There
the problem is solved in an auxiliary
domain# using a finite element method on a fairly structured
mesh which allows the use of fast solvers. The auxiliary
domain# contains the domain # and the
solution
in# is found as a solution of a distributed optimal control problem such that it satisfies
the prescribed boundary conditions of the problem in the domain #. The same idea is also used
in [9] where a least squares method is used. In [12], an embedding method is proposed in which
a combination of Fourier approximations and boundary integral equations is used. Essentially, a
Fourier approximation for a solution of the inhomogeneous equation
in# is found, and then, the
solution in # for the homogeneous equation is sought using the boundary integral methods.
# Institute of Mathematics, Romanian Academy of Sciences, P.O. Box 1-764, RO-70700 Bucharest, Romania (e-mail:
Department of Mathematics, Texas A&M University, College Station, TX-77843, USA (e-mail:
prabir.daripa@math.tamu.edu)
In recent years, progress in this field has been substantial, especially in the use of the Lagrange
multiplier techniques. In this connection, the works of Girault, Glowinski, Hesla, Joseph, Kuznetsov,
Lopez, Pan, Periaux ([14], [15], [16], [17] and [18]) should be cited.
There are many problems for which an exact solution on some particular domains may be known
or computed numerically very e#ciently. In these cases, an embedding domain method associated
with a boundary optimal control problem would allow one to find the solution of the problem
very e#ciently in an otherwise complicated domain. Specifically, the particular solution of the
inhomogeneous equation can be used to reduce the problem to solving an homogeneous equation
in # subject to appropriate conditions on the boundary of the domain #. This solution in the
complicated domain # can be obtained via an optimal boundary control problem where one seeks
for the solution of the same homogeneous problem in the auxiliary
domain# that would satisfy
the boundary conditions on the domain #. We mention that the boundary control approach has
been already used by Makinen, Neittaanmaki and Tiba for optimal shape design and two-phase
Stefan-type problems ([29], [32]). Also, recently there has been an enormous progress in shape
optimization using the fictitious domain approaches. We can cite here, for instance, the works of
Haslinger, Klarbring, Makinen, Neittaanmaki and Tiba (see [8], [21], [22], [23] and [33])
among many others.
In section 2, an optimal boundary control problem involving an elliptic equation is formulated.
In this formulation, the solution on the auxiliary
domain# is sought such that it satisfies the
boundary conditions on the domain #. In general, such an optimal control problem leads to an
illposed problem, and consequently it may not have a solution.
Using a controllability theorem of J. L. Lions, it is proved here that the solutions of the problems
in # can be approximated within any specified error, however small, by the solutions of the problems
in# for appropriate values of the boundary conditions. In section 3, it is shown that our optimal
control problem has an unique solution in a finite dimensional space. Consequently, considering
a family of finite dimensional subspaces having their union dense in the whole space of controls,
we can approximate the solution of the problem in # with the solutions of the problems
in#
using finite dimensional optimal boundary control problems. Since the values of the solutions in
are approximately calculated on the boundary of the domain #, we study the optimal control
problem with boundary observations in a finite dimensional subspace in section 4. In section 5, we
extend the results obtained for the interior problems to the exterior problems. In section 6, we give
some numerical examples for both bounded and unbounded domains. The numerical results are
presented to show the validity and high accuracy of the method. Finally, in section 7 we provide
some concluding remarks. There is still a large room for further improvement and numerical tests.
In a future work, we will apply this method in conjunction with a fast algorithm ([6], [7]) to solve
other elliptic problems in complicated domains.
Controllability
Let
(i.e. the maps defining the boundaries of the domains and their derivatives
are Lipschitz continuous) be two bounded domains in R N such that
Their boundaries are
denoted by # and #, respectively.
In this paper, we use domain embedding and optimal boundary control approach to solve the
following elliptic equation:
subject to either Dirichlet boundary conditions
or Neumann boundary conditions
#y
#nA (#)
h # on #, (2.3)
#nA (#)
is the outward conormal derivative associated with A.
We assume that the operator A is of the form
with a ij # C (1),1 (
# and there exists a constant c > 0 such that
in# for any (# 1 ,    , # N ) # R N . Also, we assume that f # L
A function y # H 1/2 (#) will be called a solution of the Dirichlet problem (2.1)-(2.2) if it verifies
equation (2.1) in the sense of distributions and the boundary conditions (2.2) in the sense of traces
in L 2 (#). A function y # H 1/2 (#) will be called a solution of the Neumann problem (2.1), (2.3) if
it verifies the equation (2.1) in the sense of distributions and the boundary conditions (2.3) in the
sense of traces in H -1 (#) (see [27], Chap. 2, 7).
The Dirichlet problem (2.1)-(2.2) has an unique solution and it depends continuously on the
data
If there exists a constant c 0 > 0 such that a 0 # c 0 in #, then the Neumann problem (2.1), (2.3) has
an unique solution and it depends continuously on the data
If a #, then the Neumann problem (2.1), (2.3) has a solution if
In this case the problem has an unique solution in H 1/2 (#)/R and
We also we remark that the solution of problem (2.1)-(2.2) can be viewed (see [27], Chap. 2,
as the solution of the problem
#n A #)
for any # H 2
and that a solution of problem (2.1), (2.3) is also solution of the problem
for any # H 2 (#
#n A #)
where A # is the adjoint operator of A given by
(a ji
Evidently, the above results also hold for problems in the
domain#
We consider in the following only the cases in which our problems have unique solutions, i.e. the
Dirichlet problems, and we assume in case of the Neumann problems that there exists a constant
such that a 0 # c 0 in #
Below we use the notations and the notions of optimal control from Lions [26]. First, we will study
the controllability of the solutions of the above two problems (defined by (2.1) through (2.3)) in #
with the solutions of a Dirichlet problem in # Let
be the space of controls. The state of the system for a control v # L 2 (#) will be given by the solution
1/2(# of the following Dirichlet problem
in#
In the case of the Dirichlet problem (2.1)-(2.2), the space of observations will be
and the cost function is given by
and y(v) is the solution of problem (2.11). For the Neumann problem given by
and (2.3), the space of observations will be
and the cost function will be
Remark 2.1 Since y(v) # H
2(#5 we have y(v) # H 2 (D) for any
domain D which satisfies
# (see [30], Chap. 4, 1.2, Theorem 1.3, for instance).
Therefore, having the same values on both the sides of #. Also, #y(v)
#nA (#)
Consequently, above two cost functions make
sense.
Proposition 2.1 A control u # L 2 (#) satisfies where the control function is given by
(2.13), if and only if the solution of (2.11) for
#nA (#)
and
where y is the solution of the Dirichlet problem defined by (2.1) and (2.2) in the domain #. The
same result holds if the control function is given by (2.15) and y is the solution of the Neumann
problem (2.1) and (2.3).
Proof. Let
1/2(# be the solution of problem (2.11) corresponding to an u # L 2 (#) such
that with the control function given by (2.13). Consequently, y(u) verifies equation (2.1)
in the sense of distributions and the boundary condition (2.2) in the sense of traces. It gives
in #. Since y(u) satisfies equation (2.11)
# in the sense of distributions, then, evidently, y(u)
is a solution the equation in (2.16). From (2.17) and Remark 2.1 we obtain that y(u) also satisfies
the two boundary conditions of (2.17). The reverse implication is evident.
The same arguments also hold for the Neumann problem defined by (2.1) and (2.3) and the
control function given by (2.15). #
Since (2.16) is not a properly posed problem, it follows from the above proposition that the optimal
control might not exist. However, J. L. Lions proves in [26] (Chap. 2, 5.3, Theorem 5.1)
a controllability theorem which can be directly applied to our problem. We mention this theorem
below.
Lions Controllability Theorem The set { #z0 (v)
is dense in H -1 (#),
#) is the solution of the problem
z
z
Now, we can easily prove
Lemma 2.1 For any g # L 2 (#), the set { #z(v)
is dense in H -1 (#)
#) is the solution of the problem
on #.
Proof. Let z # H
#) be the solution of the problem
on #.
Using z z in the Lions controllability Theorem, we get that the set { #(z(v)-z)
is dense in H -1 (#). Hence, the lemma follows. #
The following theorem proves controllability of the solutions of problems in # by the solutions of
Dirichlet problems in # In the proof of this theorem below, we use the spaces # s introduced in
Lions and Magenes [27], Chap. 2, 6.3. For the sake of completeness, we give here definitions of
these spaces # s .
Let #(x) be a function in D(
# ) which is positive
in# and vanishes on #. We also assume that
for any x 0 #, the following limit
lim
exists and is positive, where d(x, #) is the distance from x
# to the boundary #. Then, for
, the space # s is defined by
With the norm
|#s
the space #
s(# is a Hilbert space, and
D(# is dense in #
Now, for a positive non-integer real the integer part of s and 0 < # < 1, the space
# s is, as in case of the spaces H s , the intermediate space
Finally, for negative real values -s, s > 0, the space #
-s(# is the dual space of #
Theorem 2.1 The set {y(v) |# : v # L 2 (#)} is dense, using the norm of H 1/2 (#), in {y # H 1/2 (#)
f in #}, where y(v) # H
1/2(# is the solution of the Dirichlet problem (2.11) for a given v # L 2 (#).
Proof. Let us consider y # H 1/2 (#) such that Ay = f in #, and a real number # > 0. We denote
the traces of y on # by
#nA (#)
(#). From the previous lemma, it
follows that there exists v # L 2 (#) such that the solution z(v # H
-  #) of problem (2.18)
satisfies
|
#)
be the solution of the Dirichlet problem (2.11) corresponding to v # and let us define
y # y on #
#.
1/2(# and satisfies in the sense of distributions the equation
in#
and the boundary conditions
Consider, as in Remark 2.1, a fixed domain D such that
D(#2 we have
3/2(#, where C(D) depends only on the domain D. Therefore,
Taking into account the continuity of the solution on the data (see Lions and Magenes [27], Chap.
2, 7.3, Theorem 7.4), we get
Below, the controllability of the solutions of the Dirichlet and the Neumann problems (given by
(2.1),(2.2) and (2.1), (2.3) respectively) in # by Neumann problems
in# is discussed.
Now, as a set of controls we can take the space
and for a v # H -1 (#), the state of the system will be the solution y(v) # H
1/2(# of the problem
in#
= v on #. (2.20)
We remark that the following
solution of problem (2.11)} #
solution of problem (2.20)}
establish a bijective correspondence. Consequently, Proposition 2.1 also holds if the space of controls
there is changed to H -1 (#) and the states y(v) of the system are solutions of problem (2.20).
Theorem 2.1 in this case becomes
Theorem 2.2 The set {y(v) |# : v # H -1 (#)} is dense, using the norm of H 1/2 (#), in {y #
1/2(# is a solution of the Neumann problem (2.20) for a
3 Controllability with finite dimensional spaces
Let {U # be a family of finite dimensional subspaces of the space L 2 (#) such that given (2.10) as
a space of controls with the Dirichlet problems, we have
U # is dense in
For a v # L 2 (#) we consider the solution y
1/2(# of the problem
in#
We fix an U # . The cost functions J defined by (2.13) and (2.15) are di#erentiable and convex.
Consequently, an optimal control
v#U#
exists if and only if it is a solution of the equation
when the control function is (2.13), and
#nA (#) ,
#y # (v)
#y # (v)
for any v # U # , (3.5)
when the control function is (2.15). Above, y(u # ) is the solution of problem (2.11) corresponding to
(v) is the solution of problem (3.2) corresponding to v. If y f # H
2(# is the solution of
the problem
in#
then, for a v # L 2 (#), we have
where y(v) and y # (v) are the solutions of problems (2.11) and (3.2), respectively. Therefore, we can
rewrite problems (3.4) and (3.5) as
and
#nA (#)
for any v # U # ,
respectively. Next, we prove the following
Lemma 3.1 For a fixed #, let # 1 ,    , #n# , n # N be a basis of U # and y # i ) be the solution of
problem (3.2) for { #y #1 )
#nA (#y #n # )
#nA (#}
are linearly independent sets.
Proof. From Remark 2.1, we have y # (v) # H 2 (D) for any domain D which satisfies
D # and consequently, y # (v) # H 3/2 (#) for any v # L 2 (#). Assume that for # 1 ,    , # n# R we
have
and therefore y on #. This implies that
#y #1 #1++#n #n # )
From (3.10) and (3.11), we get y
#, and therefore, # 1
on #, or # The second part of the statement
can be proved using similar arguments. #
The following proposition proves the existence and uniqueness of the optimal control when the states
of the system are the solutions of the Dirichlet problems.
Proposition 3.1 Let us consider a fixed U # . Then, problems (3.8) and (3.9) have unique solutions.
Consequently, if the boundary conditions of Dirichlet problems (2.11) lie in the finite dimensional
space U # , then there exists an unique optimal control of problem (3.3) corresponding to either the
Dirichlet problem (2.1), (2.2) or the Neumann problem (2.1), (2.3).
Proof. For a given #, let V # denote the subspace of L 2 (#) generated by {y
is a basis of U # , and y # i ) is the solution of problem (3.2) with Since the norms
are equivalent to the
norm
the above lemma then implies that there exists two positive constants c
and C such that
Consequently, from the Lax-Milgram lemma we get that equation (3.8) has an unique solution. A
similar reasoning proves that equation (3.9) also has an unique solution. This time we use the norm
equivalence
#y # (v)
#) | H
in the Lax-Milgram lemma. #
The following theorem proves the controllability of the solutions of the Dirichlet and Neumann
problems in # by the solutions of the Dirichlet problems in #
Theorem 3.1 Let {U # be a family of finite dimensional spaces satisfying (3.1). We associate the
solution y of the Dirichlet problem (2.1), (2.2) in # with problem (3.3) in which the cost function
is given by (2.13). Also, the solution y of the Neumann problem (2.1), (2.3) will be associated with
problem (3.3) in which the cost function is given by (2.15). In both the cases, there exists a positive
constant C such that for any # > 0, there exists U # such that
where u # U # is the optimal control of the corresponding problem (3.3) with # , and y(u # )
is the solution of problem (2.11) with
Proof. Let us consider an # > 0 and y # H 1/2 (#) be the solution of problem (2.1), (2.2). From
Theorem 2.1, there exists v # L 2 (#) such that y(v # H
1/2(#5 the solution of problem (2.11) with
(#. Consequently, there exists a constant C 1 such that
U # is dense in L 2 (#), there exists # and v # U # such that |v # - v # | L 2 (# and then,
there exists a positive constant C 2 such that
From (3.12) and (3.13) we get
and consequently,
where u # L 2 (#) is the unique optimal control of problem (3.3) on U # with the cost function
given by (2.13). Therefore,
A similar reasoning can be made for the solution y # H 1/2 (#) of problem (2.1), (2.3). #
Using the basis # 1 ,    , #n# of the space U # we define the matrix
and the vector
Then problem (3.8) can be written as
#,1 ,    , #,n# R
Consequently, using Theorem 3.1, the solution y of problem (2.1), (2.2) can be obtained within
any prescribed error by setting the restriction to # of
where #,1 ,    , #,n# ) is the solution of algebraic system (3.16). Above, y f is the solution of
problem (3.6) and y # i ) are the solutions of problems (3.2) with
An algebraic system (3.16) is also obtained in the case of problem (3.9). This time the matrix
of the system is given by
#nA (#) ,
#nA (#)
and the free term is
#y f
#nA (#) ,
#nA (#)
Therefore, using Theorem 3.1, the solution y of problem (2.1), (2.3) can be estimated by (3.17).
Also, y f is the solution of problem (3.6) and y # i ) are the solutions of problems (3.2) with
The case of the controllability with finite dimensional optimal controls for states of the system given
by the solution of a Neumann problem will be treated in a similar way. As in the previous section
the space of the controls will be U given in (2.19), and the state of the system y(v) # H
be given by the solution of Neumann problem (2.20) for a v # H
Let {U # be a family of finite dimensional subspaces of the space H -1 (#) such that
U # is dense in
This time, the function y # (v) # H
appearing in (3.4), (3.5), (3.8) and (3.9) will be the
solution of the problem
in#
#y # (v)
= v on #, (3.21)
for a v # H
appearing in (3.7), (3.8) and (3.9) will be the solution of the
problem
in#
#y f
With these changes, Lemma 3.1 also holds in this case, and the proof of the following proposition is
similar to that of Proposition 3.1.
Proposition 3.2 For a given U # the problems (3.8) and (3.9) have unique solutions. Consequently,
if the boundary conditions of Neumann problems (2.20) lie in the finite dimensional space U # , then
there exists an unique optimal control of problem (3.3), corresponding to either Dirichlet problem
(2.1), (2.2), or Neumann problem (2.1), (2.3).
A proof similar to that given for Theorem 3.1 can also be given for the following theorem.
Theorem 3.2 Let {U # be a family of finite dimensional spaces satisfying (3.20). We associate
the solution y # H 1/2 (#) of problem (2.1), (2.2) with problem (3.3) in which the cost function is
given by (2.13). Also, the solution y of problem (2.1), (2.3) will be associated with problem (3.3) in
which the cost function is given by (2.15). In both the cases, there exists a constant C such that for
any # > 0, there exists # such that
where u # U # is the optimal control of the corresponding problem (3.3) with # , and y(u # )
is the solution of problem (2.20) with
Evidently, in the case of the controllability with solutions of Neumann problem (2.20) we can also
write algebraic systems (3.16) using a basis # 1 ,    , #n# of a given subspace U # of the space
(#). As in the case of the controllability with solutions of the Dirichlet problem (2.11), these
algebraic systems have unique solutions.
Remark 3.1 We have defined y f as a solution of problems (3.6) or (3.22) in order to have
, respectively, on the boundary #. In fact, we can replace
y(v) by y # (v) in the cost functions (2.13) and (2.15), with y f # H
satisfying only
in# , (3.23)
and the results obtained in this section still hold.
Indeed, the two sets
corresponding to y f given by
(3.23) and (3.6), y # (v) being the solution of (3.2), are identical to the set {y(v) # H
being the solution of (2.11). Also, the two sets
corresponding to y f given by (3.23) and (3.22), y # (v) being the solution of (3.21), are
identical with the set {y(v) # H
being the solution of (2.20).
4 Approximate observations in finite dimensional spaces
In practical computing, we calculate the values of y # (v) at some points on # and use in (3.8),
some interpolations of these functions. We will see below that using these interpolations,
i.e. observations in finite dimensional subspaces, we can still obtain the approximate solutions of
problems (2.1), (2.2) and (2.1), (2.3).
As in the previous sections we deal at first with the case when the states of the system will be given
by the Dirichlet problem (2.11). Let U # be a fixed finite dimensional subspace of
the basis # 1 ,    , #n# .
Let us assume that for problem (2.1), (2.2), we choose a family of finite dimensional spaces {H  }
such that
H  is dense in
Similarly, we choose the finite dimensional spaces {H  } such that
H  is dense in
for problem (2.1), (2.3). We notice that H  given in (4.1) or (4.2) is a subspace of H given in (2.12)
or (2.14), respectively.
Let us consider a fixed H , given in (4.1) or (4.2) depending on the problem we have to solve.
For a given # i , we will consider the solution y problem (3.2) corresponding to
and we will approximate its trace on # by y # ,i
. Also, the approximation of #y
#nA (#)
on # will
be denoted by #y #
#nA (#)
Since the systems (3.16) have an unique solution, the determinants of the matrices # given in
and (3.18) are non-zero. Consequently, if |y # i
,i | L 2 (#) or | #y
are
small enough, then the matrices
and
#nA (#) ,
#y #
have non-zero determinants. In this case, the algebraic systems
have unique solutions. In this system the free term is
if the matrix # is given by (4.3), and
#y f
#nA (#) ,
#y # ,i
if the matrix # is given by (4.4). Above, we have denoted by g # and h # some approximations
in H of g # and h # , respectively. Also, y f and #y f
#nA (#)
are some approximations of y f and #y f
#nA (#)
in the corresponding H of L 2 (#) and H -1 (#), respectively, with y f # H
satisfying (3.23).
If we write for a vector # 1 ,    , # n# R n# ,
and
#y #
(#)
#nA (#)
problems analogous to (3.8) and (3.9) can be written as
and
#nA (#)
(#)
#nA (#)
for any # R n# ,
whose solutions # are the optimal control for the following cost functions
J (#) =2 |y #
and
J (#) =2
#y #
(#)
#nA (#)
respectively.
The solution y of problems (2.1), (2.2) and (2.1), (2.3) can be approximated with the restriction
to # of
#,1 ,    , #,n# ) being the solution of appropriate algebraic system (4.5).
For a vector, # 1 ,    , # n# ), we will use the norm
| and the corresponding matrix
norm will be denoted by ||  ||. From (3.17) and (4.14) we have
depends only on the basis in U # . Since
and from algebraic systems (3.16) and (4.5) we have #
l # and
l # , we get that
there exists C # > 0, depending on the basis in U # , such that
In the case of matrices (3.14) and (4.3) and the free terms (3.15) and (4.6), we have
Instead, if we take matrices (3.18) and (4.4) and the free terms (3.19) and (4.7), then we get
1#i#n# |
#y # ,i
#y f
#y f
1#i#n# |
#y # ,i
where C is a constant and C # depends on the basis in U # .
In the case when the states of the system will be given by the Neumann problem (2.20), U # will
be a subspace of what we said above in the case of the Dirichlet problems
in#
can be applied in the case of the Neumann problems in # the only di#erence being that this time
are the solutions of problems (3.21) with
In both cases, when the control is e#ected via Dirichlet and Neumann problems, using Theorems
3.1 and 3.2, and equations (4.15)-(4.18), we obtain
Theorem 4.1 Let {U # be a family of finite dimensional spaces satisfying (3.1) if we consider
problem (2.11), or satisfying (3.20) if we consider problem (2.20). Also, we associate problem (2.1),
(2.2) or (2.1), (2.3) with a family of spaces {H  }  satisfying (4.1) or (4.2), respectively. Then, for
any # > 0, there exists # such that the following holds.
(i) if the space H  is taken such that |y # i
,i | L 2 (#) , are small enough, y is
the solution of problem (2.1), (2.2) and y(u # ) is given by (4.14) in which # is the solution of
algebraic system (4.5) with the matrix given in (4.3) and free term in (4.6) then the algebraic system
(4.5) has an unique solution and
,i | L 2 (# ,
(ii) if the space H  is taken such that | #y
are small enough,
y is the solution of problem (2.1), (2.3) and y(u # ) is given by (4.14) in which # is the solution
of algebraic system (4.5) with the matrix given in (4.4) and free term in (4.7) then the algebraic
system (4.5) has an unique solution and
#nA (#y f
where C is a constant and C # depends on the basis of U # .
Remark 4.1 Since the matrices # given in (4.3) and (4.4) are assumed to be non-singular, it
follows that {y # ,i } i=1,,n #
and { #y # ,i
are some linearly independent sets in L 2 (#) and
respectively. Consequently, if m  is the dimension of the corresponding subspace H  , then
5 Exterior problems
In this section, we consider the domain # R N of problems (2.1), (2.2) and (2.1), (2.3) as the
complement of the closure of a bounded domain and it lies on only one side of its boundary. The
same assumptions will be made on the
domain# of problems (2.11) and (2.20), and evidently, #
In order to follow the way in the previous sections and to prove that the solutions of the problems
in # can be approximated by the solutions of problems
in# we have to specify the spaces in which
our problems have solutions and also, their correspondence with the trace spaces.
First, we notice that the
domain# -
being bounded, then the Lions controllability Theorem does
not need to be extended to unbounded domains. Moreover, we see that the boundaries # and # of
the domains #
and# are bounded, and consequently, we can use finite open covers of them (as for
the bounded domains), to define the traces.
In order to avoid the use of the fractional spaces of the spaces in #
and# we simply remark
that since H 1/2 (#) is dense in L 2 (#), then using the continuity of the solution on the data (of the
and the continuity of the conormal derivative operator #
on the boundary #, we
get from the Lions controllability Theorem that
The set #z0 (v)
#) is the
solution of the problem
z
z
Now, we associate to the operator A the symmetric bilinear form
a(y,
#y
#z
# a 0 yz for y, z # H
which is continuous on H
1(#1 Evidently, a is also continuous on H 1 (#)  H 1 (#). Now,
and taking the boundary data g # H 1/2 (#) and h # H -1/2 (#), then problems (2.1),
(2.2) and (2.1), (2.3) can be written in the following variational form
fz for any z # H 1(#)
and
z for any z # H 1 (#), (5.2)
respectively. Similar equations can also be written for problems (2.11) and (2.20).
Therefore, if there exists a constant c 0 > 0 such that a 0 # c 0 in # then the bilinear form a is
1(#6842056# i.e. there exists a constant # > 0 such that #|y| 2
a(y, y) for any y # H
1(#5 It
follows from the Lax-Milgram lemma that problems (2.11) and (2.20) have unique weak solutions in
1(#6 Naturally, problems (2.1), (2.2) and (2.1), (2.3) in # also have unique weak solutions given
by the solutions of problems (5.1) and (5.2), respectively.
We know that there exits an isomorphism and homeomorphism of H
(see Theorem 7.53, p. 216, in [1], or Theorem 5.5, p. 99, and Theorem 5.7, p. 103, in [30]), i.e.
there are two constants such that we have
. For any y # H
1(# , there exists v # H 1/2 (#) such that
y | H
.
. For any v # H 1/2 (#), there exists y # H
1(# such that y = v on # and | y | H
Using this correspondence we can easily prove the continuous dependence of the solutions on
data. For instance, for problems (2.1), (2.2) and (2.1), (2.3) we have
and
respectively.
Therefore, if there exists a constant c 0 > 0 such that a 0 # c 0 in # then we can proceed in the
same manner and obtain similar results for the exterior problems to those obtained in the previous
sections for the interior problems. Evidently, in this case we take
as a space of the controls for problem (2.11), in place of that given in (2.10), and the space of controls
for problem (2.20) will be taken as
in place of the space given in (2.19).
If a in # the domain being unbounded, then our problems might not have solutions in the
classical Sobolev spaces (see [10]), and we have to introduce the weighted spaces which take into
account the particular behavior of the solutions to infinity.
For domains in R 2 , we use the weighted spaces introduced in [24, 25], specifically
where D
# is the space of the distributions on # and r denotes the distance from the origin. The
norm on W
1(# is given by
| v | W
(L
For domains in R N , N # 3, appropriate spaces, introduced in [20] and used in [19, 31], are
with the norm
| v | W
(L
We remark that the space H
1(# is continuously embedded in W
1(#8 and the two spaces coincide
for the bounded domains. We use W 1
to denote the closure of
D(# in W
Concerning the space of the traces of the functions in W
1(#6 we notice that the boundary #
being bounded, these traces lie in H 1/2 (#). This fact immediately follows considering a bounded
domain D
# such that #D and taking into account that W 1 (D) and H 1 (D) are identical.
Assuming that
and using the spaces W 1 in place of the spaces H 1 , we can rewrite the problems (5.1) and (5.2),
and also, similar equations for problems (2.11) and (2.20).
For 2, the bilinear form a(y, z) generates on W 1
an equivalent norm with that induced by
(see [24]). Also, the bilinear form a(y, z) generates on W
1(# /R a norm which is equivalent
to the standard norm.
For N # 3, the above introduced norm on W 1 (R N ) is equivalent to that generated by a (see
[20]). Now, if we extend the functions in W 1
0(# with zero in R N
- # we get that the bilinear form
a(y, z) also generates on W 1
a norm equivalent to that induced by W
1(#1 Moreover, using the
fact that the
domain# is the complement of a bounded set, it can be proved that the bilinear form
a(y, z) generates in W
1(# a norm equivalent to the above introduced norm.
Therefore, we can conclude that, in the case of a our exterior problems have unique
solutions in the spaces W 1 if N # 3. If 2, the Dirichlet problems have unique solutions in W 1 ,
and the Neumann problems have unique solutions in W 1 /R.
Using the fact that on the bounded domains D the spaces W 1 (D) and H 1 (D) coincide, the
continuous embedding of H
1(# in W
1(#7 and the homeomorphism and isomorphism between
we can easily prove that there exits an homeomorphism and isomorphism
between W
1(# and W
Consequently, we get the following continuous dependence on
the data of the solution y of problem (2.1), (2.2).
and
Concerning the problem (2.1), (2.3), we have
and
Therefore, we can prove in a manner similar to the previous sections that when a
and N # 3, the solutions of the Dirichlet and Neumann problems in # can be approximated with
solutions of both the Dirichlet and the Neumann problems in # Naturally, the controls will be taken
in the appropriate space (5.3) or (5.4). If a
on# and 2, the solutions of the Dirichlet
problems in # can be approximated with solutions of the Dirichlet in # the Neumann problems not
having unique solutions.
6 Numerical Results
In this section, we consider some fixed U # and H  , and we drop the subscripts # and . First, we
summarize the results obtained in the previous sections concerning the algebraic system we have to
solve to obtain the solutions, within a prescribed error, of problems (2.1), (2.2) or (2.1), (2.3) using
the solutions of problems (2.11) or (2.20).
We saw that, if, for both the bounded and unbounded domains, there exists a constant c 0 > 0
such that the coe#cient a 0 of the operator A satisfies a 0 # c 0 in # then the solutions of problems
(2.1), (2.2) or (2.1), (2.3) can be estimated by the solutions of both problems (2.11) and (2.20).
If a
in# for both the bounded and the unbounded domains, then the solutions of problems
(2.1), (2.2) can be estimated by the solutions of problems (2.11). If a the domains are
unbounded and N # 3, then the solutions of problems (2.1), (2.3) can be obtained from the solutions
of problems (2.20).
Actually, we have to solve an algebraic system (4.5) which we rewrite as
Some remarks on the computing of the elements of the matrix # and the free term l are made below.
a) Depending on the problem in # we choose the finite dimensional subspace of controls U # U .
If we use problem (2.11), U is L 2 (#)
if# is bounded and is H 1/2 (#)
if# is unbounded. Also, U is
if# is bounded and is H -1/2 (#)
if# is unbounded, if we use problem (2.20). Let # 1 ,    , #n ,
N, be the basis of U .
b) Depending on the problem
in# and the coe#cient a 0 of the operator A, we calculate the
values of y # i n, at the nodes of a mesh on #, the boundary of #. For problem (2.11),
we calculate y # i solutions of problems (3.2). For problem (2.20), if there
exists a constant c 0 > 0 such that a 0 # c 0 in # or if a
and# is unbounded, we
calculate y # i solutions of problems (3.21).
c) Using the values of y # i calculated at the nodes of the mesh in #, we will
compute the elements of the matrix # which are some inner products in we have to
solve problem (2.1), (2.2), or in we have to solve problem (2.1), (2.3). We notice
that the inner product in H -1 (#) is given by
where -# is the Laplace-Beltrami operator on # is the tangential gradient on #, I # is the
identity operator, and
Evidently, use of this inner product implies the solving of n problems of the above type to find
the corresponding of #y
#nA (#)
, and one problem to find the corresponding of h #y f
#nA (#)
. The finite
dimensional subspace H # H depends on the numerical integration method that we use. We remark
that the matrix # is symmetric and full.
d) The elements of the free term l will also be some inner products in the corresponding space
of observations H. In these inner products we use a solution y f of equation (3.23), and the data
# or h # in the boundary conditions of the problem we have to solve, (2.1), (2.2) or (2.1), (2.3),
respectively.
e) The elements of the matrix # and the free term l depend on the problems
in# and #, and
also, on the coe#cient a 0 of the operator A. For problem (2.1), (2.2), the matrix # and the free term
l are given in (4.3) and (4.6), respectively. For problem (2.1), (2.3), if there exists a constant c 0 > 0
such that a 0 # c 0 in # or if a
and# is unbounded, the matrix # and the free term
l are given in (4.4) and (4.7), respectively. Evidently, in these equations, y #
, the approximations in
H of y # i on the problem in #
Finally, if # 1 ,    , # n ) is the solution of algebraic system (6.1), and y is the solution of the
problem we have to solve, then its approximation is the restriction to # of
If we use the finite element method to calculate the functions y f and y # i
is not necessary to adapt the meshes
in# to the geometry of #. The values of these functions at
the points of # mentioned in the items b) and d) above can be found by interpolation using their
values at the mesh nodes. In our numerical examples we use an explicit formula for these functions.
Actually, we can find explicit formulae for the solutions of most problems in simple shaped domains
We saw that the matrices # given in (3.14) and (3.18) are non-singular and therefore, problems
have unique solutions. Also, algebraic systems (6.1) have unique solutions if their matrices
are good approximations in H of the matrix and the free term of the algebraic systems
(3.16), respectively. In fact, this approximation depends on the numerical integration on #. Also,
from Remark 4.1 we must take n # m, n being the dimension of U and m the dimension of H .
However, as we saw in Section 2, the problem in infinite space may not have a solution. Consequently,
for very large n, we might obtain algebraic systems (3.16) almost singular. These algebraic systems
can be solved by an iterative method, the conjugate gradient method, for instance, but we wanted
to see whether the algebraic system is non-singular and we applied the Gauss method, checking
the diagonal elements during the elimination phase. Below, we show that our numerical results are
encouraging.
Our numerical tests refer to both the interior and exterior Dirichlet problems
on #,
where # R 2 is either the interior or exterior domain of the square # centered at the origin, with
the sides parallel to the axes and of length 2 unit. The approximate solution of this problem is given
by the solutions of the Dirichlet problems
in#
in which the
domain# is either the disc # centered at the origin with radius equal to 2, or the
exterior domain of the disc # centered at the origin with radius equal to 0.99. The solutions of these
interior and exterior Dirichlet problems
in# are found by the Poisson formulae
2#r #|=r
dS # .
The square # is discretized with m equidistant points and H is taken as the space of the continuous
piecewise linear functions. The circle # is similarly discretized with n equidistant points and U is
taken as the space of the piecewise constant functions. The values of the integrals in the Poisson
equation at the points on # are calculated using the numerical integration with 3 nodes. The integrals
in the inner products in L 2 (#) are calculated by an exact formula, i.e., if we have on # two continuous
piecewise linear functions y 1 and y 2 , such that for x # [x k , x k+1
then
[y k
being the mesh size on #.
It is worth mentioning right at the outset that all computations below used fifteen significant
digits (double precision). Numerical experiments were carried out with three sets of data
for the problems in #: g #
only for
the interior problem, g # In the case of the interior problems, exact
known solutions are compared with the computed ones at 19 equidistant points on a diagonal of
the square: (-1.4,-1.4),  ,(0,0),  ,(1.4,1.4). The maximum of the relative error between exact and
computed solutions is denoted in the tables below by err d . In the case of the unbounded domains,
we do not know the exact solution but we can directly compute the error between the values of the
boundary conditions g # and the values of the computed solution given in (6.2), at the considered
points on #. The maximum of the relative error at these points is denoted in these tables by err b .
The errors err d and err b in the three examples for the interior problem are almost the same.
In

Table

6.1, we give an example for g(x 1 , x -4. In this example,
corresponding to the mesh size 0.1 on #. In all these cases, err d < err b as shown in the Table 6.1. For
these simple examples, it is not necessary to compute y f numerically as we used y f
36 .67121E-12 .11648E-06

Table

6.1. Tests for the interior Dirichlet problem.
The smaller diagonal element during the Gauss elimination method is of the order 10 -17 for
and of the order 10 -14 for It is greater than 10 -10 for
We should mention that in both cases with 72, the last pivot is of the order
However, we notice an increase in error for n > 60 (see Table 6.1), and these cases should be
cautiously considered. In all these cases the error err b , which can be calculated for any example, is
a good indicator of the computational accuracy.
In

Table

6.2, we give an example for the exterior problem with In
this example, corresponding to the mesh size of 1/15 on #.
90 0.18828E+00

Table

6.2. Tests for the exterior Dirichlet problem.
The smaller diagonal element during the Gauss elimination method is of order 10 -15 for
of the order 10 -14 for and it is greater than 10 -12 for
Conclusions
In this paper we studied, for both interior and exterior problems, the approximation of the solutions
of the Dirichlet (Neumann) problems in # with the solutions of the Dirichlet (Neumann)
in# by
means of an optimal boundary control problem
in# with observations on the boundary of the
domain #. As we saw in Section 2, such an optimal boundary control problem might lead to an
illposed problem if the space of the controls is infinite dimensional. We proved that if the controls
are taken in a finite dimensional subspace, then our problem has an unique optimal control. Using
the J. L. Lions controllability theorem we also proved that the set of the restrictions to # of the
solutions of the Dirichlet (Neumann) problems
in# is dense in the set of the solutions of the Dirichlet
(Neumann) problems in #. It is natural to take a family of finite dimensional spaces whose union is
dense in the space of the boundary conditions of the problem in # Then the set of restrictions to
# of the solutions of the Dirichlet (Neumann) problems
in# which assume values of the boundary
conditions in that union is dense in the set of solutions of these problems in #. Consequently, the
optimal boundary control problem
in# in which the controls are taken in a finite dimensional space
of such a family, will provide a solution of a Dirichlet (Neumann) problem
in# whose restriction
to # will approximate the solution of the Dirichlet (Neumann) problem in #. Actually, such an
optimal control problem whose controls are taken in a finite dimensional space leads to the solution
of a linear algebraic system. Since in the practical applications, the values of the solutions
in#
are approximately calculated on the boundary of the domain #, we also studied the optimal control
problem with boundary observations in a finite dimensional subspace.
Our primary goal in this paper has been to present this method and provide some theory and
calculations in order to bring forth some of the merits of this method as well as to provide some
theoretical support. However, we feel at this point that it is perhaps worthwhile to make some
remarks on this method against the backdrop of a somewhat di#erent technique within the same
framework: namely, Lagrange multiplier technique. For reasons mentioned below, we think that the
boundary control approach is simpler, more flexible, and can be more accurate than the Lagrange
multiplier approach to domain embedding methods.
As we mentioned earlier in the section on Introduction, very good results have been obtained
in recent years by using the Lagrange multiplier approach to the domain embedding methods. In
the Lagrange multiplier approach to domain embedding methods, values of Lagrange multipliers
are sought such that the solution of the problem in the
domain# satisfies the specified boundary
conditions of the problem in #. In fact, values of the Lagrange multipliers are essentially the jump
at the boundary # of #, in the normal derivative of the solution in # In the boundary control
approach to domain embedding methods proposed in this paper, boundary values of the solution
in# are sought such that this solution satisfies the specified boundary conditions of the problem
in #. Consequently, the numbers of supplementary unknowns introduced in the two methods are
equivalent.
In the Lagrange multipliers approach, one solves a saddle-point problem for the nodal values of
the solution
in# and those of the multipliers. In the method proposed in this paper, the problem is
reduced to solving a linear algebraic system. The construction of this linear system needs the solution
of many problems in # but these problems, being completely independent, can be simultaneously
solved on parallel machines. We point out that in the conjugate gradient method associated with
the Lagrange multipliers method, one has to solve at each iteration a problem
in# which has an
additional term arising from the Lagrange multipliers. In contrast, our method requires solutions of
simpler problems, o#ers good parallelization opportunities, and consequently a low computational
complexity.
In our approach, we solve several problems in the simple shaped
domain# and a linear combination
of these solutions provides the final desired solution. Since fast solution techniques are usually
available for many problems in simpler domains, we can expect to get very accurate solutions in our
approach. In fact, our numerical results for both, the interior and the exterior Dirichlet, problems
confirm a high accuracy of the proposed method. In the Lagrange multipliers method, problem
formulation introduces an additional term (which is an integral on #) which almost always forces
one to use the finite element method to solve for desired solutions even in regular domains. Very
good approximate solutions in the presence of such additional term usually requires additional computational
complexity such as the use of finer meshes. This increases the dimension of problems in
each iteration.
In a future work, we will apply the proposed method in conjunction with fast algorithms to
solve general elliptic partial di#erential equation in complex geometries. There we will also make
numerical comparisons between the results obtained with the method proposed in this paper and
that using the Lagrange multipliers method.

Acknowledgment

: The second author (Prabir Daripa) acknowledges the financial support of the
Texas Advanced Research Program (Grant No. TARP-97010366-030).



--R


Methods of fictitious domains for a second order elliptic equation with natural boundary conditions

Domain embedding methods for the Stokes equations
The direct solution of the discrete Poisson equation on irregular regions
A fast algorithm to solve nonhomogeneous Cauchy-Riemann equations in the complex plane
Singular Integral Transforms and Fast Numerical Algorithms: I


Les espaces du type Beppo-Levi

A spectral embedding method applied to the advection-di#usion equation

analysis of a finite element realization of a fictitious domain/domain decomposition method for elliptic problems
On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrange multiplier method
Joseph D.


On the coupling boundary integral and finite element methods for the exterior Stokes problem in 3-D
Espaces de Sobolev avec poids.


Shape optimization of materially non-linear bodies in con- tact
Equations int

Optimal control of systems governed by partial di


A boundary controllability approach in optimal shape design problems


On the approximation of the boundary control in two-phase Stefan-type problems
An embedding of domain approach in free boundary problems and optimal design
Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions
On the numerical solution of Helmholtz equation by the capacitance matrix method
A locally refined finite rectangular grid finite element method.
--TR
optimal control;domain embedding methods