Internat. J. Math.
Vol. 10 No. 2 (1987)
&Math. Sci.
345
345-360
AN H -GALERKIN METHOD FOR A STEFAN PROBLEM WITH A
QUASILINEAR PARABOLIC EQUATION IN NON-DIVERGENCE FORM
A.K. PANI
and
P.C. DAS
Department of Mathematics
I.I.T., KANPUR
208016, India
(Received December 10, 1985)
ABSTRACT.
L
Optimal error estimates in
2
H
H2_no rms
and
are established
for a
single phase Stefan problem with quasilinear parabolic equation in non-divergence form
by an
HI-Galerkin
procedure.
H1-Galerkin
KEY NDDS AND PHRASES.
procedure, finite element approximation, nonlinear
Stefan problem, non-divergence form, error analysis.
1980 AMS SUBJECT CLASSIFICATION CODE. 65N15, 65N30.
I.
INTRDDION.
With the help of Galerkin finite element methods, Nitsche in his pioneering works
[I]-[3] established error estimates for linear problems, proposed earlier by Magenes
[4]. We extended his analysis to nonlinear problems in divergence form [5]-[6]. In
the present work, a single phase Stefan problem with quasilinear parabolic equation in
form
non-divergence
is
under
and
considered
2
estimates for Galerkin approximation in L
We
require
more
regularity
H
for
assumptions
conditions
appropriate
H2
as well as
the
one
present
than
[5]-[6], and consequently we improve upon the estimates in
discussed in
The organization of the paper is as follows:
optimal
error
norms are established
for
the
cases
L2-norm.
In section 2, the description of
the problem and the transformed system with some preliminaries are presented.
weak formulation and
deals
with
error
optimal
HI-Galerkin
in
estimates
are
approximations
procedure are discussed in section 3.
projection and
an auxiliary
established,
L
2
H
some
and
assuming
approximation Lemmas.
H2-norms
for
existence
of
Section 4
In section 5
time
Galerkin
approximate
solution.
continuous
the
The
Finally, in section 6 the question of global existence and uniqueness of the Galerkln
approximation is discussed.
2.
PIOBLEM DESCRIPTION AND D0t4AIN FIXINg.
The nonlinear heat conduction with change of phase can be modelled as a single
phase
nonlinear
{y- (0,S(T))
and
Stefan
S(T)
problem in a variable domain
(T) x (0,T ),
O
known to be the free boundary
{U,S}, U
U (y,T), S
S(T)
such that
R(T)
We state this problem as
follows:
Find a pair
where
U
satisfies
346
A.K. PANI and P.C. DAS
a(U)
U
Uyy --0,
for
(y,T) e (T)
(0, T O
(2.1)
with initial and boundary conditions
g(y),
U(y,0)
u y (0,.)
for y e I
(2.2)
(0,I)
0
for >0
>
for
U(S() ,)
and
(2.3)
S, the free boundary satisfies
-Uy(S(),),
S
with
0
0,
S(0)
I.
>
for
(2.4)
0
The above problems is a special case of the general situation discus-
sed in Fasano et. al. [7], where
’a’
=-
depends only on U, q
0
-U (S(T),)
Y
and
in their notations.
We use the following notiations.
Let
I
(u,v)
uv dx
()
Hm(())
and
lull
Let
2
be the usual Sobolev space
()
(u,u).
For
Wm’P((T)),
be a bounded domain for
R
each
for
nonnegative integer
2
p
>
0.
m, let
with the norm
dE,
Further,
wm"((1))
In
case
If
X
I
is defined as usual with the norm
(),
we shall omi
I
from
be a normed linear space with norm
Hm(I)
I1.11 X
L (I)
and
and
W
#: (a,b)
denote by
<q<(R)
and
I*l
is accordingly defined.
m
(I)
X,
and norm
then we
STEFAN PROBLEM WITH A QUASILINEAR PARABOLIC EQUATION
(a b)
In case
(0 T)
for
m
X
and
H
or
x
use
we
nience,
Throughout this work,
K
-
Wm’(R)
xx 2;
x
x’
m
W
or
2
t
’
347
Jl Jl
we write simply
I*llwk
for
and (I)
,,,,wk,q(Hm
For conve-
wTM
(l,t),
(x,t).
if
On occasion, we will
will always denote a generic constant
show that a constant depends on certain parameters, while independent of others
a(.), g
We shall now state our main assumption on
and the solution
U,S
and
call them collectively ’condition B’.
CONDITION B.
(i)
For
p E R, a(p)
(ii)
For
p
E
apl’ lapp[
(iii)
>
,
and there is a common bound
[apppl KI"
and
The initial function
condition that is
is a positive constant.
where
c3(R)
R, a(p)
gy(0)
K!
>
0
such that
lal,
is sufficiently smooth and satisfies the compatibility
g
g(1)
0.
The problem (2.1)-(2.4) has a unique solution.
(iv)
For the existence and uniqueness of the solution of (2.1)-(2.4), see Fasano et.al. [7].
ing regularity condition.
Let
U
e
S
W
For an integer
(0,T0; Hr+l (fl(T))
W
2
r
of (2.1)-(2.4) satisfies the follow-
U,S
Further it is assumed that the solution
>
I,
(0,r0; Hr+! ((T)))
n
W
, (0,T0;
I’(R)(0,T O ).
’
(())),
be the bound for the functions in above mentioned
K
2
We fix the free boundary, using Landau type transformation [8]
s-l()y,
x
>
(2.5)
0.
Further, we introduce an additional transformation in time scale given by
I0 S-2(T ’)d’,
t(T)
t
(2.6)
in order to decouple the resulting transformed system.
that the function
u(x,t)
u
t
A routine calculation shows
U(y,) satisfies
a(U)Uxx -Ux(1)x Ux,
u(x,0)
g(x), x
Ux(0,t)
u(l,t)
x e I, t
(2.7)
(2.8)
I;
0, t
(0,T]
>
0
(2.9)
348
A.K. PANI and P.C. DAS
and the function
S(1)
s(t)
ds
dt
with
s(0)
Here,
U,S
t
satisfies
Ux(1)s,
>
t
(2.10)
0
I.
T
corresponds to
T
I
are carried over to
Note that all the regularity assumptions for
O
u,s
with the bound say
R
assumptions are collectively called
and the new regularity
K
2
Further, the integral (2.6) can be rewritten
I.
as
dr
s
d--{
3.
2
(t),
HI-GALKIIN
WAK IeORILATION AND
(0)
with
(2.11)
0.
PROKDURK.
Consider the space:
0
H2(1)
H2 (I):
{v
v (0)
x
0}
v(1)
The weak formulation of (2.7)-(2.9) is given by
0
+
(Utx,V x)
with
u(x,0)
HI-Galerkin
(a(U)Uxx, Vxx)
v e
H2(1)
and
>
t
0
(3.1)
g(x).
Let
Procedure.
0
H2(1)
Ux(1)(xu x, Vxx),
0
S
(0
h
< h_<
I)
be
a
finite
dimensional
subspace
of
0r,2
belonging to regular
S
family, for a definition see Oden et. al. [9] and
h
satisfying the following approximation and inverse properties:
(i)
For
v E
0
H2(1),
Hm(1)
there is a constant
IIv-xIIj
inf
Xe
<
K^
independent of
h
such that
for j
0
2 and 2<m<r+l;
llVllm .....
K hm-j
0
h
0
Now we call
uh:
(0,T]
0
S
h
an
(uhtx’ Xx) + (a(uh)
HI-Galerkin
h
Uxx’ Xxx)
approximation of
h. l)(XUx’h Xxx),
u
x
u,
if it satisfies
X
0
S
h
(3.2)
and the initial condition
uh(x,0)
Qhg(x),
(3.3)
0
where
Qh
Further,
given by
is an appropriate projection of
the
Galerkin
approximations
u
s
h
onto S
and
h
h
at t
of
s
0,
and
to be defined later.
respectively
are
STEFAN PROBLEM WITH A QUASILINEAR PARABOLIC EQUATION
dSh
uh(1)s
h
x
dt
with
Sh(0)
with
h(0)
349
(3.4)
and
dlh
2
d--{4.
Sh’
(3.5)
0.
SOME APPROXIILTION LE.
Set
type inequality for
The boundedness and Garding
arguments.
LEMMA 4.!.
For
u
W
for u e W I’
Ux(1)(xv x ,Wxx );
(a(U)Vxx Wxx)
A(u;v w)
’
A
v and w e H
2
(4.1)
can be established by standard
0
and
v
H2ljt
w
(4.2)
and
llVxxll
_>
where
M, a
p
and
M
are constants, but
and
p
may depend on
u
Define
A(u;v,w) +
A(u;v,w)
0
A(u,.,.)
Note that
>_
Let
u
0
S
H
is coercive in
Ao(u;
.
L
O(Vx,Wx )"
2,
that is
llVxxll
(4.4)
be an approximation of
h
x
u-u,X)
0,
u
Ap:
with respect to the form
0
S h,
X
(4.5)
Now, an application of Lax-Milgram theorem shows the existence of
a unique solution u
of equation (4.5).
Consider
0
L* (u)
For
e
L2(1),
define
L*(u)
(a(U)xx) xx
e H4
;
0
2
H
by
x
I
XXlx=l xXXlx=0
/
Ux( l)(Xxx
x
Oxx
u e H
2
(4.6)
(4.7)
0.
350
A.K. PANI and P.C. DAS
0
H2(1)
v e
Then, for
we get
%(u;v,).
(v, L*(u))
D(L*)
Thus, defining
as
{
D(L*)
we
from
have
D(L*)
the
C
depends on
O
u
rivatives
u
2:
(I)
e L2
O}
xxx(0)
A
that at least a weak solution
exists and the regularity
It11,
(4.9)
and its derivatives, holds.
We now need to obtain some estimates of
u.
for our future use.
qt’
xx
and boundedness of
positivity
of (4.7) for each
Let
0
H
e H4
11’114 < Co
where
(4.8)
our purpose.
and its temporal de-
The following Lemma proves very convenient for
0
LEIA 4.2.
H2(1)
Let
and satisfy
0
Ao(u;h ,X)
F(X),
e
(4.10)
Sh,
0
F:
where
H2(1)
R
Let there exist constants M
and linear.
such that
IF(*)I < M1
II*xxtl’ *
and
M
2
with M
>
M
2
0
H2
(4.11)
and
(4.12)
Then, for sufficiently small
h
(4.13)
0
X Sh
and
It’ ]l <--K3
[(M
+ inf
e
where
K
3
PROOF.
K3(a,O,M,C0;K0)
0
S
[1 -XII2)h 2 + M2]
(4.14)
h
is used as generic constant.
Note that
Ap(u; ,)
Ap(u;,-X)
By coercive property (4.4) for
l xxl
_<
AO,
F(-X) + F(h),
0
e S
h.
we get
I< II* II
+
inf
X
0
S
h
II -II _
+
II x IIl-
STEFAN PROBLEM WITH A QUASlLINEAR PARABOLIC EQUATION
351
0
I1,11 _< <>-[ inf I1
0
S
and the estimate (4.13)
Aubin-Nitsche
e
For
Multiply both the sides of (4.7) by
(,)
-
+ 2KM
L2-estimate,
In order to get an
follows.
arguments.
duality
h
t1,
L2(1),
define
to obtain for
0
H
u
we follow here
e D(L*)
by
(4.7).
2,
A (u;,)
( ,L* (u))
A (u;,@-X) + F(- @) + F(@)
(4.15)
From (4.9), (4.13) and (4.15), we obtain the required estimate (4.14).
The next Lemma contains the error estimates related to
4.3.
For
t E
[o,T],
and
t
the following estimates
(4.16)
and
I1,,:11. <_. % ’<llull + Ilu,:llm >,
and
0,1,2
j
2
<_ m
(4.17)
r+1,
Here K
are positive constants depending on parameters expressed
and K
4
5
through the following expressions that is
hold.
K
4
PROOF.
K4(K ,K
0
Put
3)
and
5(K0 ,K
K
and
K
5
F
lu: II ,- "’’ lull W
W
in the previous Lemma 4.2 to get
0
Xg S
,K 3 ,K 4
h
0
X
0
S
h
.)-
352
A.K. PANI and P.C. DAS
0
Similarly, we get the estimate (4.16), for j
0, consequently, the estimate for
follows from the interpolation inequality,
IIBIII
I/2
In order to estimate
we differentiate (4.5) with respect to
t’
ida(u) nxx, Xxx)
(-
Ap(u;nt,X)
+
Utx(1)
(xn x,
’t’
and obtain
(4.18)
Xxx).
0
Identifying the right hand side of (4.18) with
where
K
6
depends on
K
0
e D(L*)
Further, for
F()
and
u e H
2,
@
H2(1)
-
lu ll W
and
F(X), we see that for
we get on integration by parts
(rx ’(at (U)xx)x) -Utx (1)(n (Xqxx)x)
(,(a t(u)#xx)xx) +
Utx(1)(,(X#xx) x)
and
where
K
7
K
7
( ,llu xxl L., Iluxxll.L
the desired estimate (4.17)
interpolation inequality.
LK 4.4.
PROOF.
for
j
and
Ilu ll W
0,2.
For j
We shall also need later the following estimate for
There is a constant
Define an auxiliary function
L*(u)
Cxxx x=O
Cxx Ix=l
I, as usual we make use of the
O,
O;
1.
x
e I
(a,K0,
#
H
4
M; K4) such that for 2
0
2
H
as a solution of
m
Bx(1).
r+l.
STEFAN PROBLEM WITH A QUASILINEAR PARABOLIC EQUATION
n
Multiplying by
353
the first equation and integrating by parts, we obtain
0
<
AC;n,
sh
-x),
h
Hence, the result follows.
5. A PRIORI ERROR ESTDLATES FOR CoIrlhJOUS TINE GALEIIN APPROKINATION.
such that a Galerkin approximation
[[uhl[.
K( H2
where
uh(x,
0)
uh( x,
Let
u
m
>
u
and
e
K*,
u
in (3.2) exists and satisfies,
h
<
0
for
h
< h0,
(5.1)
satisfying
X)
u
Suppose
Then,
0
S
0, X
(5.2)
h-
.
h
h,
the Galerkln approx-
given as in (5.2).
Further, assume that
satisfies (4.5) and
u-
(3.2) with
Qh
u
(, , K*,
there is a constant
K I,
K4,
K and K
5
8)
such
4
IEXlIL=(L2)
PROOF.
u
Qhg,
Qhg,
is defined by
u
(5.1) holds.
that for
g-
0
e S
h
u(x, 0).
0)
h
THORR 5.1.
imation of
<
is defined as
Ao(g;
Clearly,
K* and h0
this section, we assume that there are positive constants
Throughout
+
II;xxl[ L2(L 2) <__ K9 (l[utll L 2 <Hm)
From (4.5) and (3.1) with
(Utx Xx)
+
AD(u;
v
X,
I[ull L2(hm) ).
(5.3)
we get
-(ntx, X x)
u, X)
+
+ P(uX,
Xx),
X
0
S
h.
Subtracting this from (3.2), we obtain
(tx’ Xx)
+
A0(u
h
u
h
X)
Ap
(u; u, X)
+
p (
(nix, X x)
x x)
-P(n x, X x)
(5.4)
A.K. PANI and P.C. DAS
354
But
Ap
(u
h
u
h
Ap
X)
+ ([a(u
h)
a
,
X
+
(x’
xx
Ux(l)(x x
Xxx)
(u)] u
x (1)(xuxh, xx
From (5.4)-(5.5) with
a(uh xx’ Xxx
(u; u, )
+
h
nx(1)(XUx X.xx
x )"
(5.5)
it follows on integrating by parts with respect to
the two terms on the right hand side of (5.4),
d
2 dt
IIxl12
h
+ (a(u)
+ ([a(u)
a(.)
Using
>
d
a
(n
xx
xx
(uh)]
u
,
(5.1) and replacing
2
2
XX
u
xx
t
xx
by
+
(r
xx
xx
-nx(1)(xu
n,
u
Ux(1)(Xx xx)
)+
+
X
(1)(xu
XX
).
we obtain
(5.6)
Since
I=x()l
i
IIxtl ’z II=xxll ’z o =
2
last term and the inequality ab
in
(5.6),
]III
we get using
, Itxll
d
Now
With
with
this
appropriate
choice
of
2
+
’
a
--+
llxll
L
b
2
for
IIxxll 2 <-.’:,o
choice of
,
e
.,
0
we get
Gronwall’s inequality the following
E,
KI0()
applying Young’s inequality for the
0; e
a, b
>
0
for the remaining terms
0
2
H
II,,xll
can be made less than or equal to
by integrating with respect
to
’t’
/2.
and using
STEFAN PROBLEM WITH A QUASILINEAR PARABOLIC EQUATION
From
2(m-2)
and (4.19) with
(4.|6)-(4.17)
3
m and 2m
355
that is
m
4,
m
we get
the desired estimate (5.3).
COROLLAR 5.2.
finite
dimensional
KII
constant
Let
all
the assumtions
0
S
subspace
and K
depending on
the
satisfy
h
the
of
r +
I1=tl
From the estimate (5.3) and
there
is
a
L (Hm)
=
I xll
<
Then
4,
m
L (Hm)
PROOF.
theorem hold and the
property.
inverse
such that for
0
previous
L
0
,
get
H m)
H m)
L
0
By inverse property for
we have
Sh,
0
L
Hence the result (5.7).
L
From Theorem 5.1, Corollary 5.2, Lemma 4.3 and triangle in-
equality we get the following theorem.
THKORE 5.3.
R
condition
K* (K*
exists
>
2K
in
Let
Further,
2)
I
the
(0,T] for 0
KI2 KI2(K4, KII
<
h
(2.7)-(2.9)
are
and
and consequently,
KI2
K1
K2).
<
=
positive
h
u
h
r+l-j
0
S
h
regularity
and
o
(5 8)
0 1,2
j
Besides, for sufficiently small
2K
2
<
and
h
0,1,2
j
Corollary 5.2 and Lemma 4.3 by triangle inequality.
llull L
>
3,
K*.
H
2
+
are immediate from the Theorem 5.1,
To prove (5.9), we note
llell L
H
2
<-- K2 + KI2 hr-1
< 2,
r
(5.9)
K*
can be choosen independent of
<
constants
>
The estimates (5.8) for
L(
the
satisfy
of (3.2) satisfying (5.1)
h
3,
Then, the following estimates hold for r
< h0.
<_
lu ll L( H2
PROOF.
there
that
such that an approximate solution
I1 1
where
of
u
solution
suppose
for sufficiently small h and r
>
3.
356
A.K. PANI and P.C. DAS
U
We are now looking for approximations of
solution of (2.1)-(2.4).
S, where the pair {U, S} is the
and
U
The Galerkin approximations
uh(y,T)
sh(T)
h
and S
h
are given by
uh(x,t)
(5.10)
Sh(t
(5.11)
where,
y
T =T
and
Sh,
T
h
h.
(3.4), (3.5) respectively.
are defined by
TRKORKM 5.4.
(5.12)
sh(t)x
the
that
Suppose
B
condition
and the regularity condition
Then the following estimates hold for
are satisfied.
[[S-Sh[
0 (h
r
+
r
>
R
3,
I)
(5.13)
e (0,T0
(5.14)
L (0,T
0
and
0 (h r+l-j), j
(0,T0; HJ((T)))
L
(5.15)
0,I,2,
is interpreted as
where
L
fl (T)
with
PROOF.
(0,T0; HJ((T)))
(0,
min
(S(T),
HJ((T))
0
aT
Sh(T))).
From (2.10) and (3.4), we have
t
t
Is- l
An
application
m
r +
of
_<S (l"x<’>l
0
Gronwall’s
/
/
inequality
and
the
S0
estimates
Is-s, l
(4.19),
(5.3),
for
gives
< K(K2’
<_ El3 hr+l
){h2(r-l)lul L(2 Hr+l
for
r
>
3.
+
llxxll L(2 L 2
(5.16)
STEFAN PROBLEM WITH A QUASILINEAR PARABOLIC EQUATION
The estimate (5.13) is immediate from
L (O,T
(5.16),
0)
L (0,T)
Further, the estimate (5.14) follows from (2.11), (3.5) and (5.16).
(0,T0; HJ(()))
L
357
if we note that
Finally since
HJ(1))
L (0,T;
we obtain the required estimate (5.15).
6.
GLOBAL EXISTENCE AND HNIQHEESS OF THE GALERKIN APPROXIMATION.
Now we consider the problem of existence of the Galerkin approximation
the domain of existence of
Ao(u
h
h
u
h
in
Towards this, let us recall (5.4) and note
u.
Ap(U;
)
u
u, X)
,
Ap(u;
) + ([a (u
+
nx(1)(xuhx
x
-P(nx’
Xxx)-
h)
a(u)]
h
Uxx,
xx
x (l)(xuhx, Xxx).
From the above, we get
(tx’ Xx)
+
Ap(U;
h
Uxx,
,
(tx’
)
h
Xxx)
x
+
P(x’
h)
Ba (u-e)
Replacing
u
h
by
u-e
U
f
e
+
Ap(U;
,r
0
+
a
(u-e)
uu
(x(
x
by
E(x, t), where
({tx’ Xx)
+
Ap(U;
,[
0
+
aa
-(n t,
{, X)
.
(6.2)
E
(r1-{)d{
Xxx
+ P (n,
+ p
Xxx
(Uxx-exx) Xx x)
(x’ Xx)
-rl
x(1) (X(Ux-ex) Xxx)
+
P(x’ Xx)
Ux-ex )’ Xxx )"
0
e
e d
0
(6.1)
(6.1) with (6.2), we have
in
({tx’ Xx)
Substitute
-a(uh)]
h
+
a
+ ([a(u)
x (1)(xux, xx ).
nx(1)(xux Xxx)
But,
a(u) -a(u
x
E
H
{, X)
(u-E)
2.
Then we get
-(n t,
Xxx)
+ p(n,
Xxx)
(n-)d(Uxx-Exx), Xxx)-nx(1)(X(Ux-Ex), Xxx)
x(1)(x(u-E
x) Xxx)
X
(6.3)
which is a linear ordinary differential equation in
_
(x, 0)
The
0
E
H
2.
(6.4)
0
=
(0,T].
in the interval
(6.3)
equation
Since
e
n-,
e
n
E(x,t)
Therefore, for any E
of (6.3) with
there exists a unique solution
E
.
A.K. PANI and P.C. DAS
358
an
defines
operator
J
such that
therefore
Aj
(E),
To show the existence of a solution
u
h
(6 5)
of (3.2), we need to show that the operator
In other words, we are looking for an
equation (6.5) has a fixed point.
for each
0
2
H
E
for each
(E),
e(E)
such
that
e(E)
E
Suppose that the finite dimensional space
TH.ORM 6.I.
is the unique solution of
property and
u
conditions R
be satisfied.
of (3.2) satisfying
PROOF.
in
2 dt
Ix(1)l
Using
and
ab
a2
<-+
satisfies inverse
Further, let the regularity
there exists a solution
u
h
0
S
h
flex iI + ; ll.xll
2
b
0,
h
(6.3) to get
a,b
>
O;
< K14 ()
Choosing
<
H2(1))-
t=xll / I=xll /=,
<
.
>
Then for some
lu-uhll e (O,To;
Set
(2.7)-(2.9).
0
S
>
for the remaining terms, we get
0
Ixl
appropriately so that
applying Young’s inequality for the last term
z
2
/
c,
2, ;
K14(),
there after applying Gronwall’s inequality, we get
)tll’l 12
/
z-
I’11 L
integrating with respect to
’t’ and
STEFAN PROBLEM WITH A QUASILINEAR PARABOLIC EQUATION
359
t
)If
0
L (H
{ll +ll
2)
From the estimates (4.16), (4.17) and (4.19), it follows that
(6.6)
L(
where
)--
KI5 KI5
L
L
L
L
(6.7)
L (H)
L
I II.L H 2
+o
<+
and from (4.16), (6.6), (6.7), we get
lell L’( H2
<
K
6
Therefore, for sufficiently small
h
r-I
KI6 El6 (K
where
K
5’
4
K0;
6 ).
h
II II
L(
Now, an application of Schauder’s fixed point theorem guarantees the existence of an
The
e
such that
E
E, which is a solution of the operator equation (6.5).
uniqueness of the approximate solution
u
h
is easy to prove.
So we formalize the
above in the form of a Theorem.
TEOREM 6.2.
K
>
0.
Then
there
{llu-uhll L (H2)-< K},
of
the Theorem 6.1 be satisfied and let
all
the
hypotheses
exists
one
and only one solution
Let
for sufficiently small
h
and
u
r
>
h
E
3.
0
S
h
of (3.2) in the ball
A.K. PANI and P.C. DAS
360
I.
NITSCHE, J.A.
2.
NITSCHE, J.A.
3.
NITSCHE, J.A.
4.
Boston, London and Melbourne (1983).
MAGENES, E. Topics in Parabolic Equations: Some Typical Free Boundary Problems,
Finite Element Approximations to the One Dimensional Stefan Problem
Proc. on Recent Advances in Numerical Analysis, C. de Boor and G. Golub eds.,
Academic Press, NY (1978), 119-142.
A Finite Element Method for Parabolic Free Boundary Problems, Inst.
Naz. di Alta Matematica, Vol. I (M. Magenes ed.) (1980), 277-318.
Finite Element Approximation to One Phase Stefan Problem Free
Boundary Problems: Theory and Appl. an Interdisciplinary Symp., Motecantini,
(Italy) Vol. II (Research Notes in Math. 79), Pitman Adv. Publ. Program.
Boundary Value Problems for Linear Partial Differential Equations, Proc. NATO
Adv. Stud. Inst. Liege, Belgium, Sept. 1976, H.G. Garnier ed., D. Reidel Publ.
Co., Dortrecht (1976), 239-312.
5.
DAS, P.C. and PANI, A.K.
6.
DAS, P.C. and PANI, A.K.
7.
FASANO, A.
8.
(1979), 247-273.
LANDAU, H.A. Heat Conduction
H2-norms
for Galerkin
A Priori Error Estimates in H and
Approximations to a Single Phase Nonlinear Stefan Problem, (Submitted).
A Galerkin Method For a Single Phase Nonlinear Stefan
Problem with Dirichlet Boundary Conditions, (Submitted).
and PRIMICERIO, M. Free-Boundary Problems For Nonlinear Parabolic
Equations With Nonlinear Free Boundary Conditions, J. Math. Anal. Appl. 72
in
Melting Solid,
Quart. J. Appl.
Math.
8_. (1950),
81-94
9.
ODEN, J.T. and REDDY, J.N. An Introduction To the Mathematical Theory of Finite
Elements, Wiley Interscience, NY (1976).