UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII
2005
DIFFERENCE METHOD FOR AN ELLIPTIC SYSTEM OF
NON-LINEAR DIFFERENTIAL-FUNCTIONAL EQUATIONS
by Ryszard Mosurski
Abstract. We consider a weakly coupled system of second order differential-functional equations of elliptic type with boundary conditions of the
Dirichlet type. The system is in the general nonlinear form
fl (x, u (x) , (ul )x (x) , (ul )xx (x) , u) = 0,
where l = 1, . . . , p. We propose an implicite difference scheme for this
problem and under certain assumptions we show the convergence, stability, existence and uniqueness of the solution for this scheme. An error
estimations are also given.
1. Introduction. Let D = [0, X]n ⊂ Rn , X < +∞. We consider the
system of second order differential-functional equations of elliptic type
(1.1)
fl (x, u (x) , (ul )x (x) , (ul )xx (x) , u) = 0
for x ∈ intD (l = 1, . . . , p)
with boundary conditions of the Dirichlet type
ul (x) = ϕl (x) for x ∈ ∂D (l = 1, . . . , p) ,
∂ul
∂ul
where x = (x1 , . . . , xn ), u = u1 , . . . , up , (ul )x = ∂x
,
.
.
.
,
∂xn , (ul )xx =
1
(1.2)
2
ul
[ ∂x∂ i ∂x
]i,j=1,...,n (l = 1, . . . , p) .
j
The general form of (1.1) admits strong nonlinearity in all arguments of
fl (l = 1, . . . , p). It is known that assumptions under which we can obtain
convergence theorems for nonlinear equations of form (1.1) depend mostly on
the second order difference expressions used. In [1], [6]–[10], a seven-point
scheme is used, in [3] – the nine-point scheme and the other one is used in
1991 Mathematics Subject Classification. 65M06, 65M12, 65M15.
Key words and phrases. Difference method, elliptic system, differential-functional
equations.
182
[4]–[5]. Each time the scheme forces different assumptions about left side
of differential equation(s). More discussion on the subject can be found for
example in [5].
In our paper we use difference expressions from [4]–[5] and generalize
the results of J. Kaczmarczyk in [5], where one elliptic differential equation
was considered, to nonlinear system of elliptic differential equations which
may contain functional parts. We define an implicit difference scheme for
problem (1.1), (1.2) and under certain assumptions concerning the functions
fl , ul (l = 1, . . . , p) and the mesh size h we show the stability, uniqueness and
convergence of the solution to this scheme.
We use the well-known notation introduced in papers [1]–[10] and numerous others.
2. Assumptions A. We assume that
(A1) Scalar functions fl : E 3 (x, y, q, w, z) −→ fl (x, y, q, w, z) ∈ R
2
(l = 1, . . . , p), where
E := D × Rp × Rn × Rn × B(D), x = (x1 , . . . , xn ),
y = y1 , . . . , y p , q = (q1 , . . . , qn ), w = [wij ]i,j=1..n ,
B(D) :={z = (z1 , z2 , . . . , zp ) : ∃
∃
k∈N D1 ,...,Dk
D1 ∪ . . . ∪ Dk = D,
Di ∩ Dj = ∅ for i 6= j and restriction zl |Di ∈ C(Di )
(l = 1, . . . , p; i, j = 1, . . . , k)}
(all functions piecewise continuous on D), are such that
(2.1a)
fl (x, y, q, w, z) − fl (x, ȳ, q̄, w̄, z) =
p
X
αlµ (yµ − ȳµ ) +
µ=1
(2.1b)
n
X
βli (qi − q̄i ) +
i=1
n
X
γlij (wij − w̄ij ) ,
i,j=1
|fl (x, y, q, w, z) − fl (x, y, q, w, z̄)| ≤ χl ||z − z̄|| (l = 1, . . . , p)
2
for any x ∈ D, y, ȳ ∈ Rp , q, q̄ ∈ Rn , w, w̄ ∈ Rn , z, z̄ ∈ B(D).
Functions αlµ , βli , γlij are defined in E, χl are constants (l, µ = 1, . . . , p;
i, j = 1, . . . , n) and
(2.2)
||z|| := max {sup |zl (x)|} for z ∈ B(D).
l=1,...,p x∈D
(A2) There exist constants Γ, L such that functions αlµ , βli , γlij (l, µ = 1, . . . , p;
i, j = 1, . . . , n) satisfy in E following conditions
(2.3)
0 ≤ αlµ (l 6= µ),
αll + χl +
p
X
αlµ ≤ −L < 0
µ=1
µ6=l
(2.4)
|βli | ≤ Γ.
(l = 1, . . . , p),
183
There exist bounded in E and symmetric matrices [Glij (x)]i,j=1..n (l =
1, . . . , p) and constant g such that
(2.5)
γlii (x, y, q, w, z) −
n
X
Glij (x) ≥ g > 0 , |γlij | ≤ Glij (i 6= j),
i,j=1
i6=j
(2.6)
γlij = γlji in E (i, j = 1, . . . , n; j = 1, . . . , p).
(A3) There exists a solution u(x) of problem (1.1), (1.2) such that u ∈ C 2 (D).
3. Discretization. We introduce the uniform net in the cube D. If a
sequence of indices M = (m1 , . . . , mn ), mi = 0, 1, . . . , N (i = 1, . . . , n) is
given, then by xM we denote the nodal point with the coordinates xM =
mi
mn
1
(xm
1 , . . . , xn ), where xi = mi h (i = 1, . . . , n) , 0 < h = X/N and N ≥ 2.
Let
Z := {M : 0 ≤ mi ≤ N, i = 1, . . . , n},
(3.1)
Z1 := {M : 1 ≤ mi ≤ N, i = 1, . . . , n},
Z2 := {M : 0 ≤ mi ≤ N − 1, i = 1, . . . , n},
and
−i(M ) := (m1 , . . . ,mi−1 , mi − 1, mi+1 , . . . , mn )
(M ∈ Z1 ),
i(M ) := (m1 , . . . ,mi−1 , mi + 1, mi+1 , . . . , mn )
(M ∈ Z2 )
(i = 1, . . . , n).
For any net function al : Z 3 M −→ aM
l ∈ R (l = 1, . . . , p), define the
following operators:
i(M )
−i(M )
i
−1
=
0.5h
a
−
a
,
(3.2)
aM
l
l
l
(3.3)
ij
aM
l

h−2 ai(M ) − 2aM + a−i(M )
l
l
l
=
0.25h−2 ai(j(M )) − a−i(j(M )) − ai(
l
l
l
for i = j,
−j(M ))
+
−i( −j(M ))
al
(M ∈ Z1 ∩ Z2 ; i, j = 1, . . . , n) .
+
aM
:=
l
(3.4)
n
X
i(j(M ))
−2
GM
· (al
lij 0.25h
−i(j(M ))
+ al
i,j=1
i6=j
−i( −j(M ))
+al
i(M )
− 4al
−i(M )
− 4al
i( −j(M ))
+ al
+ 4aM
l ).
for i 6= j,
184
Every function b = (b1 , . . . , bp ) ∈ B(D) is approximated by b∗ = b∗1 , . . . , b∗p ∈
B(D), where
X
(3.5)
b∗µ (x) :=
θM (x)bM
µ ,
M ∈Z
(3.6)
(
0
θM (x) :=
1
for x ∈
/ IM ,
for x ∈ IM ,
IM := {x ∈ D : mi h ≤ xi < (mi + 1) h, i = 1, . . . , n} ,
(3.7)
M
bM
µ :=bµ (x )
(µ = 1, . . . , p; M ∈ Z) .
In the same way, for every net function c: Z 3 M → cM ∈ Rp , we define
as
X
(3.8)
c∗µ (x) :=
θM (x)cM
(µ = 1, . . . , p).
µ
c∗
M ∈Z
For differential-functional system (1.1), (1.2), let v M (M ∈ Z) be a solution
of the system of difference equations
(3.9) fl xM , v M , vlM I , vlM IJ , v ∗ + vlM + = 0, (M ∈ Z1 ∩ Z2 ; l = 1, . . . , p)
with the boundary conditions
(3.10)
M
vlM = ϕM
l := ϕl (x ), (M ∈ Z r (Z1 ∩ Z2 ); l = 1, . . . , p),
where v M = (vµM )µ=1..p , vlM I = (vlM i )i=1..n , vlM IJ = [vlM ij ]i,j=1..n , (l = 1, . . . , p).
The operators vlM i , vlM ij , vlM + (l = 1, . . . , p; i, j = 1, . . . , n) and the function v ∗ are defined by (3.2)–(3.8).
System (3.9) and the boundary conditions (3.10) are generated by system
(1.1) with the boundary conditions (1.2) with the terms vlM + (l = 1, . . . , p)
added. Consider also a “perturbed” difference system
fl xM , wM , wlM I , wlM IJ , w∗ + wlM + = εM
l (h),
(3.9’)
(M ∈ Z1 ∩ Z2 ; l = 1, . . . , p)
with the boundary conditions
(3.10’)
wlM = ϕM
l , (M ∈ Z r (Z1 ∩ Z2 ); l = 1, . . . , p).
Let v and w be the solutions of problems (3.9)–(3.10) and (3.9’)–(3.10’), respectively. Define
(3.11)
rM := wM − v M = (rµM )µ=1..p
(M ∈ Z).
Then
(3.12)
rµM i =wµM i − vµM i ,
rµM ij = wµM ij − vµM ij ,
rµ∗ = wµ∗ − vµ∗
(µ = 1, . . . , p; i, j = 1, . . . , n; M ∈ Z1 ∩ Z2 ).
185
4. Stability theorem.
Theorem 1 (Stability). If the mesh size h satisfies the inequality
− 0.5Γh + g ≥ 0
(4.1)
and assumptions (A1)–(A2) hold then the following estimation is true:
|rlM | ≤
(4.2)
ε(h)
L
(M ∈ Z; l = 1, . . . , p),
where
(4.3)
ε(h) :=
max |εM
l (h)|.
M ∈Z1 ∩Z2
l=1,...,p
That means that difference scheme (3.9), (3.10) is stable.
Proof. Since sets Z and {1, .., p} are finite, there exist A ∈ Z and k ∈
{1, .., p} such that
|rkA | = max |rlM |,
(4.4)
M ∈Z
l=1,...,p
where r is the function defined by (3.11).
If A ∈ Z r(Z1 ∩Z2 ), then rkA = 0, because of the boundary conditions (3.10)
and (3.10’). In this case, (4.2) follows. Thus, we may assume A ∈ Z1 ∩ Z2 .
From (3.9), (3.9’), (2.1a), (2.1b) and formulas (3.12) we derive
(4.5)
M IJ
M
M
MI
, w∗ + wlM +
εM
l (h) =fl x , w , wl , wl
=fl xM , wM , wlM I , wlM IJ , w∗ − fl xM , v M , vlM I , vlM IJ , v ∗ + rlM +
=fl (xM , wM , wlM I , wlM IJ , w∗ ) − fl (xM , v M , vlM I , vlM IJ , w∗ )
+ fl (xM , v M , vlM I , vlM IJ , w∗ ) − fl (xM , v M , vlM I , vlM IJ , v ∗ ) + rlM +
=
p
X
M M
αlµ
rµ +
µ=1
n
X
i=1
M
+ fl (x , v
M
n
X
βliM rlM i +
M M ij
γlij
rl
+ rlM +
i,j=1
, vlM L , vlM IJ , w∗ )
− fl (xM , v M , vlM I , vlM IJ , v ∗ ) = M
l (h)
and
p
X
µ=1
p
X
µ=1
M M
αlµ
rµ +
n
X
βliM rlM i +
i=1
M M
αlµ
rµ +
n
X
i=1
n
X
M M ij
γlij
rl
+ rlM + − χl ||r∗ || ≤ M
l (h),
i,j=1
βliM rlM i +
n
X
i,j=1
M M ij
γlij
rl
+ rlM + + χl ||r∗ || ≥ M
l (h).
186
By (2.2) there is
(4.6)
||r∗ || = max {sup |
X
l=1,...,p x∈D
M ∈Z
θM (x)rlM |} = max {max|rlM |} = |rkA |,
l=1,...,p M ∈Z
which leads to inequalities
p
X
(4.7)
µ=1
p
X
M M
αlµ
rµ
+
n
X
βliM rlM i
+
i=1
M M
rµ +
αlµ
µ=1
n
X
n
X
M M ij
γlij
rl
+ rlM + −χl |rkA | ≤ M
l (h),
i,j=1
βliM rlM i +
i=1
n
X
M M ij
+ rlM + +χl |rkA | ≥ M
rl
γlij
l (h)
i,j=1
(M ∈ Z1 ∩ Z2 ; l = 1, . . . , p).
For M = A and l = k, from (4.7) and definition (4.2), we obtain the
following two inequalities:
(4.8)
p
X
A A
rµ
αkµ
+
µ=1
(4.9)
p
X
µ=1
n
X
A Ai
rk
βki
+
+
n
X
A Ai
βki
rk
i=1
A Aij
rk + rkA+ − χk |rkA | ≤ (h),
γkij
i,j=1
i=1
A A
αkµ
rµ
n
X
+
n
X
A Aij
rk + rkA+ + χk |rkA | ≥ −(h).
γkij
i,j=1
i) Assume rkA ≥ 0. Then
(4.10)
|rkA | = rkA and rlM ≤ rkA (l = 1..p; M ∈ Z).
Now we shall prove that
(4.11)
n
X
i=1
A Ai
βki
rk +
n
X
i,j=1
A Aij
γkij
rk + rkA+ ≤ 0.
187
We proceed as follows:
n
X
=
A Ai
βki
rk +
i=1
n
X
n
X
A Aij
γkij
rk + rlA+
i,j=1
i(A)
A
βki
0.5h−1 (rk
−
−i(A)
rk
)
n
X
+
i=1
+
+
i(A)
A −2
h (rk
γkii
−i(A)
− 2rkA + rk
)
i=1
n
X
i,j=1
i6=j
n
X
i(j(A))
A
γkij
0.25h−2 (rk
−2
GA
kij 0.25h
−i(j(A))
− rk
i(−j(A))
− rk
−i(j(A))
i(j(A))
rk
+ rk
−i(−j(A))
+ rk
i(−j(A))
+ rk
)
−i(−j(A))
+ rk
i,j=1
i6=j
i(A)
4rk
−
=
n
X
i(A)
A
βki
0.5h−1 (rk
− rkA ) +
i=1
+
+
+
+
+
+
n
X
n
X
−i(A)
4rk
−
+
4rkA
−i(A)
A
βki
0.5h−1 (rkA − rk
)
i=1
i(A)
A −2
h (rk
γkii
i=1
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
− rkA ) +
−i(A)
A −2
h (rk
γkii
− rkA )
i=1
i(j(A))
A
0.25h−2 (rk
γkij
−
rkA )
i(−j(A))
A
γkij
0.25h−2 (rkA − rk
i(j(A))
−2
GA
kij 0.25h (rk
+
−i(j(A))
A
0.25h−2 (rkA − rk
γkij
i,j=1
i6=j
n
X
−i(−j(A))
A
0.25h−2 (rk
γkij
)+
− rkA ) +
i(−j(A))
−2
GA
kij 0.25h (rk
n
X
i,j=1
i6=j
n
X
−i(j(A))
−2
GA
kij 0.25h (rk
i,j=1
i6=j
n
X
− rkA ) +
i,j=1
i6=j
i(A)
−2 A
GA
kij h (rk − rk
)+
n
X
i,j=1
i6=j
− rkA )
− rkA )
−i(−j(A))
−2
GA
kij 0.25h (rk
−i(A)
−2 A
GA
kij h (rk − rk
)
)
− rkA )
188
=
n
X
i(A)
A
A
[0.5h−1 βki
+ h−2 γkii
](rk
− rkA )
i=1
+
+
+
+
+
+
=h
n
X
−i(A)
A
A
[−0.5h−1 βki
+ h−2 γkii
](rk
i=1
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i(A)
−2 A
GA
kij h (rk − rk
n
X
)+
− rkA )
−i(A)
−2 A
GA
kij h (rk − rk
i,j=1
i6=j
i(j(A))
A
0.25h−2 [GA
kij + γkij ](rk
− rkA )
−i(j(A))
− rkA )
i(−j(A))
− rkA )
A
0.25h−2 [GA
kij − γkij ](rk
A
0.25h−2 [GA
kij − γkij ](rk
−i(−j(A))
A
0.25h−2 [GA
kij + γkij ](rk
i,j=1
i6=j
n
X
−2
i(A)
A
A
](rk
+ γkii
[0.5hβki
− rkA )
− rkA ) + h−2
+
+
+
+
+
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
−i(A)
A
A
](rk
+ γkii
[−0.5hβki
i=1
i=1
n
X
)
−2 A
GA
kij h (rk −
i(A)
rk )
n
X
+
−i(A)
−2 A
GA
kij h (rk − rk
i,j=1
i6=j
i(j(A))
A
0.25h−2 [GA
kij + γkij ](rk
− rkA )
−i(j(A))
− rkA )
i(−j(A))
− rkA )
A
0.25h−2 [GA
kij − γkij ](rk
A
0.25h−2 [GA
kij − γkij ](rk
−i(−j(A))
A
0.25h−2 [GA
kij + γkij ](rk
− rkA )
)
− rkA )
189
= h−2
(4.12)
n
X
A
A
[0.5hβki
+ γkii
−
i=1
+h
−2
n
X
n
X
+
+
+
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
− rkA )
i,j=1
i6=j
A
[0.5hβki
+
A
γkii
−
i=1
+
i(A)
GA
kij ](rk
n
X
−i(A)
GA
kij ](rk
− rkA )
i,j=1
i6=j
i(j(A))
A
0.25h−2 [GA
kij + γkij ](rk
− rkA )
−i(j(A))
− rkA )
i(−j(A))
− rkA )
A
0.25h−2 [GA
kij − γkij ](rk
A
0.25h−2 [GA
kij − γkij ](rk
−i(−j(A))
A
0.25h−2 [GA
kij + γkij ](rk
− rkA ).
i,j=1
i6=j
Each term in the square brackets is nonnegative (due to (2.4)–(2.6),
(4.1)) and terms in the parentheses are nonpositive (due to (4.10)). Thus
(4.11) follows.
From (4.9) and (4.11) we derive
(4.13)
−ε(h) ≤
p
X
µ=1
≤
p
X
A A
A A
αkµ
rµ + χk rkA = αkk
rk +
A A
αkk
rk
+
p
X
A A
rµ +χk rkA
αkµ
µ=1
µ6=k
A A
αkµ
rk +χk rkA
=
A
(αkk
+ χk +
µ=1
µ6=k
p
X
µ=1
µ6=k
due to (4.10) and (2.3). Hence, −ε(h) ≤ −LrkA and
(4.14)
|rkA | = rkA ≤
ε(h)
.
L
A
αkµ
)rkA ≤ −LrkA
190
ii) Assume rkA ≤ 0. Then |rkA | = −rkA and rlM ≥ rkA (l = 1, . . . , p; M ∈ Z).
Inequality (4.11) now takes the form
n
X
(4.15)
A Ai
βki
rk +
i=1
n
X
A Aij
γkij
rk + rkA+ ≥ 0.
i,j=1
We can see that calculations which were used to prove (4.11) may be
repeated and in (4.12) each term in the square brackets and parentheses
is nonnegative. Thus (4.15) follows. By (4.8) there is
ε(h) ≥
p
X
A A
A A
rk +
rµ − χk (−rkA ) = αkk
αkµ
≥
A A
rµ +χk rkA
αkµ
µ=1
µ6=k
µ=1
A A
rk
αkk
p
X
+
p
X
A A
rk +χk rkA
αkµ
=
A
(αkk
+ χk +
p
X
A
)rkA ≥ −LrkA .
αkµ
µ=1
µ6=k
µ=1
µ6=k
Hence ε(h) ≥ −LrkA and
(4.16)
|rkA | = −rkA ≤
ε(h)
.
L
From (4.14), (4.16), there always follows (4.2), which completes the proof
of Theorem 1.
5. Existence and uniqueness of difference solution.
Remark 5.1. Notice that (4.12) is true not only at certain nodal points
as in the proof of Theorem 1. Let z M = (zµM )µ=1..p (M ∈ Z) be an arbitrary
net function. Then, for l = 1, . . . , p; M ∈ Z1 ∩ Z2 , equality (4.12) holds true
191
in the set E :
n
n
X
X
M M ij
γlij
zl
+ zlM +
βliM zkM i +
i=1
i,j=1
= h−2
n
X
M
[0.5hβliM + γlii
−
i=1
+ h−2
n
X
n
X
+
+
+
=h
n
X
M
[0.5hβliM + γlii
−
+ h−2
− 4h
− zlM )
i(−j(M ))
− zlM )
−i(−j(M ))
[0.5hβliM
+
M
γlii
−
n
X
− zlM )
i(M )
GM
lij ]zl
j=1
j6=i
n
X
[0.5hβliM
+
M
γlii
i=1
+h
−i(j(M ))
M
0.25h−2 [GM
lij + γlij ](zl
i=1
−2
− zlM )
M
0.25h−2 [GM
lij − γlij ](zl
i,j=1
i6=j
n
X
−2
− zlM )
j=1
j6=i
M
0.25h−2 [GM
lij − γlij ](zl
i,j=1
i6=j
n
X
+h
−i(M )
GM
lij ](zl
i(j(M ))
i,j=1
i6=j
n
X
−2
n
X
M
0.25h−2 [GM
lij + γlij ](zl
i,j=1
i6=j
n
X
− zlM )
j=1
j6=i
i=1
+
i(M )
GM
lij ](zl
n
X
i,j=1
i6=j
n
X
i,j=1
i6=j
n
X
−2
i,j=1
i6=j
−
n
X
−i(M )
GM
lij ]zl
j=1
j6=i
0.25[GM
lij
+
M i(j(M ))
γlij
]zl
+h
i(−j(M ))
M
0.25[GM
lij − γlij ]zl
−2
+h
n
X
−i(j(M ))
M
0.25[GM
lij − γlij ]zl
i,j=1
i6=j
n
X
−2
−i(−j(M ))
M
0.25[GM
lij + γlij ]zl
i,j=1
i6=j
M
−2
0.25GM
lij zl − 2h
n
X
i=1
M
[0.5hβliM + γlii
−
n
X
j=1
j6=i
M
GM
lij ]zl .
192
Let
δliM
:=
0.5hβliM
+
M
γlii
−
(5.1)
n
X
GM
lij ,
δlM
:=
j=1
j6=i
n
X
δliM ,
ρM
l
:=
i=1
n
X
GM
lij
i,j=1
i6=j
(l = 1, . . . , p; i = 1, . . . , n).
Then
(5.2)
n
n
X
X
M M ij
βliM zkM i +
γlij
zl
+ zlM +
i=1
i,j=1
= h−2
n
X
i(M )
δliM zl
i=1
+h
−2
0.25
+ h−2 0.25
+ h−2
n
X
−i(M )
δliM zl
+ h−2 0.25
i=1
n
X
i,j=1
i6=j
n
X
−i(j(M ))
M
[GM
lij − γlij ]zl
+ h−2 0.25
n
X
i,j=1
i6=j
n
X
i(j(M ))
M
[GM
lij + γlij ]zl
i(−j(M ))
M
[GM
lij − γlij ]zl
i,j=1
i6=j
−i(−j(M ))
M
[GM
lij + γlij ]zl
M
−2
− 4h−2 0.25ρM
l zl − 2h
i,j=1
i6=j
n
X
δliM zlM .
i=1
Theorem 2. Let assumptions (A1)–(A2) are satisfied, there exist constants L1 , K such that inequalities
(5.3)
− L1 ≤ αll , γlii ≤ K (l = 1, . . . , p; i = 1, . . . , n)
hold true in the set E and h > 0 is so small that
(5.4)
− 0.5Γh + g ≥ 0.
Then there exists exactly one solution of differential problem (3.9)–(3.10).
Proof. Let q be the cardinality of Z, Rq×p := {V |V = (vlM )
M ∈Z }
l=1,...,p
h satisfy (5.4). Notice that from (2.3), (2.5) there follows
(5.5)
− L1 ≤ −L < 0,
K ≥ 0.
Define
(5.6)
F : Rq×p 3 V −→ F (V ) := C ∈ Rq×p ,
C = (cM
l ),
M ∈ Z, l = 1, . . . , p,
and
193
where
cM
l
(5.7)
(
fl xM , v M , vlM I , vlM IJ , v ∗ + vlM +
:=
−LvlM
for M ∈ Z1 ∩ Z2 ,
for M ∈ Z r Z1 ∩ Z2
(l = 1, . . . , p),
s
H := 1 − h L
(5.8)
and constant s is such that
1 − hs (L1 + h−2 (hnΓ + 2nK)) > 0.
(5.9)
Let Φ : Rq×p 3 V −→ Φ(V ) := hs F (V ) + V ∈ Rq×p . We shall show that
||Φ(V ) − Φ(W )||1 ≤ H||V − W ||1 for all V, W ∈ Rq×p ,
(5.10)
where ||V ||1 := max |vlM |, (V ∈ Rq×p ).
M ∈Z
l=1,...,p
There is H < 1 and
H := 1 − hs L ≥ 1 − hs L1 ≥ 1 − hs (L1 + h−2 (hnΓ + 2nK)) > 0.
So
H ∈ (0, 1).
(5.11)
Let us take two arbitrary elements V, W ∈ Rq×p , W = (wlM )
M ∈Z ,
l=1,...,p
(vlM )
M ∈Z
l=1,...,p
and let
D := Φ(V ) − Φ(W ) = (dM
l )
Y := W − V = (ylM )
M ∈Z
l=1,...,p
=
M ∈Z ,
l=1,...,p
(wlM − vlM ) M ∈Z .
l=1,...,p
Then, for M ∈ Z r Z1 ∩ Z2 , there is
s
M
M
s
M
M
dM
l = [h (−Lwl ) + wl ] − [h (−Lvl ) + vl ]
= [hs (−LylM ) + ylM ] = [−hs L + 1]ylM = HylM ≤ H||Y ||1 ,
hence
(5.12)
M
dM
l = Hyl ≤ H||Y ||1 .
For M ∈ Z1 ∩ Z2 there follows
V =
194
s
M
M
MI
M IJ
dM
, w∗ )−fl (xM , v M , vlM I , vlM IJ , v ∗ ) + ylM + ] + ylM
l = h [fl (x , w , wl , wl
= hs [fl (xM , wM , wlM I , wlM IJ , w∗ ) − fl (xM , v M , vlM I , vlM IJ , w∗ ) + ylM + ]
+ hs [fl (xM , v M , vlM I , vlM IJ , w∗ ) − fl (xM , v M , vlM I , vlM IJ , v ∗ )] + ylM
= hs [
p
X
M M
αlµ
yµ +
µ=1
n
X
βliM ylM i +
i=1
s
M
+ h [fl (x , v
M
n
X
M M ij
γlij
yl
+ ylM + ]
i,j=1
, vlM L , vlM IJ , w∗ )
− fl (xM , v M , vlM I , vlM IJ , v ∗ )] + ylM ,
and
(5.13a)
p
n
n
X
X
X
M M
M M ij
yµ − hs χl ||y ∗ || + ylM ≤ dM
αlµ
+ ylM + ] + hs
yl
γlij
hs [
βliM ylM i +
l ,
µ=1
i,j=1
i=1
(5.13b)
p
n
n
X
X
X
M M
M M ij
yµ + hs χl ||y ∗ || + ylM ≥ dM
αlµ
+ ylM + ] + hs
yl
γlij
hs [
βliM ylM i +
l ,
µ=1
i,j=1
i=1
where ||y ∗ || = max {sup |
l=1,...,p x∈D
θM (x)yµM |} = max |ylM | ≤ ||Y ||1 .
P
l=1,...,p
M ∈Z
M ∈Z
By (5.2), from (5.13b) there follows:
p
n
n
X
X
X
M+
s
M Mi
M M ij
s
M M
dM
≤
h
[
β
y
+
γ
y
+
y
]
+
h
αlµ
yµ + hs χl ||y ∗ || + ylM
k
li l
lij l
l
= hs−2
i=1
n
X
i(M )
[δliM ]yl
+ hs−2
i=1
+h
s−2
µ=1
i,j=1
n
X
−i(M )
[δliM ]yl
i=1
n
X
0.25
i,j=1
i6=j
n
X
+ hs−2 0.25
[GM
lij
+
M i(j(M ))
γlij
]yl
+h
i(−j(M ))
M
[GM
lij − γlij ]yl
s−2
0.25
n
X
i,j=1
i6=j
n
X
+ hs−2 0.25
−i(−j(M ))
M
[GM
lij + γlij ]yl
i,j=1
i6=j
i,j=1
i6=j
M
s−2
− 4hs−2 0.25[ρM
l ]yl − 2h
−i(j(M ))
M
[GM
lij − γlij ]yl
n
X
i=1
[δli ]ylM + hs
p
X
µ=1
M M
αlµ
yµ + hs χl ||y ∗ || + ylM .
195
From (A2) there follows that each term in the square brackets is nonnegative, hence
n
n
n
X
X
X
M
s−2
M
s−2
M
M
s−2
M
dl ≤ {2h
δli + h 0.25
[Glij + γlij ] + h 0.25
[GM
lij − γlij ]
i=1
i,j=1
i6=j
+ hs−2 0.25
n
X
i,j=1
i6=j
M
s−2
[GM
0.25
lij − γlij ] + h
i,j=1
i6=j
n
X
M
[GM
lij + γlij ]}||Y ||1
i,j=1
i6=j
s−2
− {4hs−2 0.25ρM
l +2h
n
X
δliM }ylM + hs αllM ylM
i=1
+ hs
p
X
M M
αlµ
yµ + hs χl ||Y ||1 + ylM
µ=1
µ6=l
≤
{2hs−2 δlM
+h
s−2
0.25
n
X
GM
lij
+h
s−2
0.25
n
X
s−2
GM
0.25
lij + h
i,j=1
i6=j
GM
lij
i,j=1
i6=j
i,j=1
i6=j
+ hs−2 0.25
n
X
n
X
GM
lij }||Y ||1
i,j=1
i6=j
s−2 M
δl }ylM + hs αllM ylM + hs
− {hs−2 ρM
l +2h
p
X
M M
yµ + hs χl ||Y ||1 + ylM
αlµ
µ=1
µ6=l
s−2 M
ρl +2hs−2 δlM }ylM + hs αllM ylM
≤ {2hs−2 δlM + hs−2 ρM
l }||Y ||1 − {h
+ hs (
p
X
M
αlµ
+ χl )||Y ||1 + ylM
µ=1
µ6=l
s−2 M
≤ {2hs−2 δlM + hs−2 ρM
ρl +2hs−2 δlM }ylM + hs αllM ylM + ylM
l }||Y ||1 − {h
s
+ h (χl +
p
X
M
αlµ
)||Y ||1
µ=1
µ6=l
s−2 M
≤ {2hs−2 δlM + hs−2 ρM
ρl −2hs−2 δlM + hs αllM + 1}ylM
l }||Y ||1 + {−h
+ hs (χl +
p
X
µ=1
µ6=l
M
αlµ
)||Y ||1
196
s−2 M
≤ {2hs−2 δlM + hs−2 ρM
ρl −2hs−2 δlM
l }||Y ||1 + | − h
+ hs αllM + 1|||Y ||1 + hs (χl +
p
X
M
αlµ
)||Y ||1
µ=1
µ6=l
≤
[2hs−2 δlM
+
hs−2 ρM
l
s
+ h (χl +
p
X
M
αlµ
)
µ=1
µ6=l
s−2 M
+ |1 + hs αllM − hs−2 ρM
δl |]||Y ||1 .
l −2h
From (5.4) and (A2) there follows ρM
l ≥ 0, and by (5.3) there is
s−2 M
1 + hs αllM − hs−2 ρM
δl
l −2h
M
s−2 M
δl ≥ 1 − hs L1 − 2hs−2 (ρM
≥ 1 + hs αllM − 2hs−2 ρM
l + δl )
l −2h
= 1 − hs L1 − 2hs−2 (
s
= 1 − h L1 − 2h
s−2
n
X
GM
lij +
i,j=1
i6=j
n
X
n
X
M
[0.5hβliM + γlii
−
i=1
[0.5hβliM
+
n
X
GM
lij ])
j=1
j6=i
M
]
γlii
s
≥1 − h L1 − 2h
s−2
n
X
(0.5hΓ + K)
i=1
i=1
= 1 − hs L1 − 2hs−2 (0.5hnΓ + nK) =1 − hs (L1 + h−2 (hnΓ + 2nK)) > 0
because of (5.9). So, from (2.3) there follows:
dM
l
≤
[2hs−2 δlM
+
hs−2 ρM
l
s
+ h (χl +
p
X
M
)
αlµ
µ=1
µ6=l
s−2 M
δl ]||Y ||1
+ 1 + hs αllM − hs−2 ρM
l −2h
s
= [h (χl +
p
X
M
)
αlµ
+1+
hs αllM ||Y
||1 = [1 + h
s
(αllM
+ χl +
p
X
M
)||Y ||1
αlµ
µ=1
µ6=l
µ=1
µ6=l
≤ [1 − hs L]||Y ||1 = H||Y ||1 .
We conclude that
(5.14)
dM
l ≤ H||Y ||1 for M ∈ Z, l = 1, . . . , p.
Similarly one can show that
(5.15)
dM
l ≥ −H||Y ||1 for M ∈ Z, l = 1, . . . , p.
So |dM
l | ≤ H||Y ||1 for M ∈ Z, l = 1, . . . , p. That means (5.10) is true.
197
From (5.11) and the Banach theorem follows existence and uniqueness of
solution to differential problem (3.9)–(3.10). This ends our proof.
6. Convergence theorem.
Theorem 3 (Convergence). Let u be a solution of system (1.1)–(1.2), u ∈
C 2 (D),
(6.1)
+
I
M IJ
ηlM (h) := fl xM , uM , uM
, u ∗ + uM
(M ∈ Z1 ∩ Z2 ; l = 1, . . . , p),
l , ul
l
η(h) :=
max |ηlM (h)|,
M ∈Z1 ∩Z2
M
uM
l := ul (x )
(M ∈ Z; l = 1, . . . , p),
l=1,...,p
v be a solution of the system (3.9)–(3.10) and assumptions (A1)–(A2) and
(4.1) hold. Then
(6.2)
M
max |uM
l − vl | ≤
M ∈Z
l=1,...,p
η(h)
L
and
(6.3)
limη(h) = 0.
h→0
This means that difference solution v of (3.9)–(3.10) converges to the differential solution u of (1.1)–(1.2).
M
Proof. For rM := uM − v M = (rµM )µ=1...p and εM
l (h) := ηl (h)
(M ∈ Z1 ∩ Z2 ) , from Theorem 1 inequality (6.2) follows.
+
−→
The proof of (6.3) is similar to that in [5]. We only show that uM
l
0 when h −→ 0 (M ∈ Z1 ∩ Z2 ; l = 1, . . . , p). Using Taylor polynomials for ul
(for simplicity we omit lower index l in u) we obtain:
1 M 2
M
M
2
ui(j(M )) = uM + uM
xi h + uxj h + ux2i h + uxi xj h +
2
1 M 2
M
M
2
u−i(j(M )) = uM − uM
xi h + uxj h + ux2i h − uxi xj h +
2
1 M 2
M
M
2
ui(−j(M )) = uM + uM
xi h − uxj h + ux2i h − uxi xj h +
2
1 M 2
M
M
2
u−i(−j(M )) = uM − uM
xi h − uxj h + ux2i h + uxi xj h +
2
1 M 2
+ M 2
ui(M ) = uM + uM
xi h + ux2i h + δij (x )h ,
2
1 M 2
− M 2
u−i(M ) = uM − uM
xi h + ux2i h + δij (x )h ,
2
1 M 2
++ M 2
u 2 h + δij
(x )h ,
2 xj
1 M 2
−+ M 2
u 2 h + δij
(x )h ,
2 xj
1 M 2
+− M 2
u 2 h + δij
(x )h ,
2 xj
1 M 2
−− M 2
u 2 h + δij
(x )h ,
2 xj
198
M
M (M ∈ Z ∩ Z )
where uM = u(xM ), uM
1
2
xi = uxi (x ) and so on, and for all x
and i, j = 1, . . . , n there exists function δ(h) such that
++ M
−+ M
+− M
−− M
|δij
(x )| ≤ δ(h), |δij
(x )| ≤ δ(h), |δij
(x )| ≤ δ(h), |δij
(x )| ≤ δ(h),
+ M
− M
|δij
(x )| ≤ δ(h), |δij
(x )| ≤ δ(h)
and δ(h) −→ 0 as h −→ 0. Now we return to lower index l:
+
uM
=
l
n
X
i(j(M ))
−2
GM
ul
lij 0.25h
−i(j(M ))
+ ul
i,j=1
i6=j
i(−j(M ))
+ ul
=
n
X
−i(−j(M ))
+ ul
i(M )
− 4ul
−i(M )
− 4ul
+ 4uM
l
−2
M
M 2
M 2
M
GM
lij 0.25h [4u + 2(ul )x2 h + 2(ul )x2 h − 8u
i
i,j=1
i6=j
j
++ M 2
−+ M 2
+− M 2
2
M
− 4(ul )M
x2 h + 4u + δij (x )h + δij (x )h + δij (x )h
i
=
−− M 2
+ M 2
− M 2
+ δij
(x )h − 4δij
(x )h − 4δij
(x )h ]
n
n
X
X
++ M
M
M
M
0.25GM
0.25Glij [−2(ul )x2 + 2(ul )x2 ] +
lij [δij (x )
i
j
i,j=1
i,j=1
i6=j
i6=j
−+ M
+− M
−− M
+ M
− M
+ δij
(x ) + δij
(x ) + δij
(x ) − 4δij
(x ) − 4δij
(x )].
Matrices Gl (x) are bounded and symmetric (l = 1, . . . , p), so the second
sum tends to 0 while h −→ 0 and the first sum is equal 0:
n
X
M
M
0.25GM
lij [−2(ul )x2 + 2(ul )x2 ]
i
i,j=1
i6=j
=−
n
X
i,j=1
i6=j
j
M
0.5GM
lij [(ul )x2 ] +
i
n
X
i,j=1
i6=j
M
0.5GM
lij (ul )x2 ] = 0.
j
This ends our proof.
References
1. Bożek B., Convergence and stability of difference scheme for an elliptic system of nonlinear differential-functional equations with boundary conditions of Dirichlet type, Ann.
Polon. Math., 44 (1984), 273–279.
199
2. Bożek B., Mosurski R., A difference scheme for an elliptic system of non-linear
differential-functional equations with Dirichlet type boundary conditions. The existence
and uniqueness of solution, Ann. Polon. Math., 44 (1984), 155–161.
3. Fitzke A., Method of difference inequalities for parabolic equations with mixed derivatives,
Ann. Polon. Math., 31 (1975), 121–129.
4. Kaczmarczyk J., The difference method for nonlinear parabolic equations, Univ. Iagel.
Acta Math., 28 (1991), 93–99.
5. Kaczmarczyk J., The difference method for nonlinear eliptic differential equations with
mixed derivative, Ann. Polon. Math., 42 (1983), 125–138.
6. Malec M., Schema des differences finies pour une equation non lineaire partielle du type
elliptique avec derive mixtes, Ann. Polon. Math., 34 (1977), 119–123.
7. Malec M., Sur une methode des differences finies pour une equation non lineaire differentielle fonctionnelle avec derive mixtes, Ann. Polon. Math., 36 (1979), 1–10.
8. Malec M., Systeme d’inegalites aux differences finies du type elliptique, Ann. Polon.
Math., 33 (1977), 235–239.
9. Mosurski R., On the stability of difference schemes for nonlinear elliptic differential equations with boundary conditions of Dirichlet type, Ann. Polon. Math., 43 (1983), 159–166.
10. Sapa L., A finite-difference method for a non-linear parabolic-elliptic system with Dirichlet
conditions, Univ. Iagel. Acta Math., 37 (1999), 363–376.
Received
January 11, 2005
AGH University of Science and Technology
Faculty of Mathematics
al. Mickiewicza 30
30-059 Kraków
Poland
e-mail : mosursk@wms.mat.agh.edu.pl