Numerical solution of nonlinear elliptic systems by block monotone iterations

Numerical solution of nonlinear elliptic systems by block monotone iterations

M. Al-Sultani

I. Boglaev

(Received 7 February 2019; revised 4 June 2019)

Abstract

We present numerical methods for solving a coupled system of nonlinear elliptic problems, where reaction functions are quasimonotone nondecreasing. We utilize block monotone iterative methods based on the Jacobi and Gauss–Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of solutions are discussed. Numerical experiments are presented.

Contents

1 Introduction C80

doi:10.21914/anziamj.v60i0.13986gives this article, c Austral. Mathematical Soc.

2019. Published July 12, 2019, as part of the Proceedings of the 18th Biennial Compu- tational Techniques and Applications Conference . issn 1445-8810. (Print two pages per sheet of paper.) Copies of this article must not be made otherwise available on the internet;

instead link directly to the doi for this article.

2 Block monotone iterative methods C81 2.1 Convergent analysis . . . . C88 2.2 Uniqueness of a solution . . . . C90

3 Numerical experiments C91

References C93

1 Introduction

Several problems in the chemical, physical and engineering sciences are characterized by coupled systems of nonlinear elliptic equations [3]. In this article, we construct block monotone iterative methods for solving the coupled system of nonlinear elliptic equations

−L

u

(x, y) + f

(x, y, u) = 0 , (x, y) ∈ ω , α = 1, 2 , (1) ω = {(x, y) : 0 < x < 1 , 0 < y < 1} ,

u(x, y) = g(x, y) , (x, y) ∈ ∂ω ,

where u = (u

, u

) , f = (f

, f

) , g = (g

, g

) , and ∂ω is the boundary of ω.

The differential operators L

, α = 1, 2 , are defined by L

u

(x, y) ≡ ε

(u

+ u

) ,

where ε

with α = 1, 2 , are positive constants. It is assumed that the functions f

and g

, α = 1, 2 , are smooth in their respective domains.

Block monotone iterative methods, based on the method of upper and lower

solutions, have only been used for solving nonlinear scalar elliptic equations [1,

2, 4]. The basic idea of the block monotone iterative methods is to decompose

a two dimensional problem into a series of one dimensional two-point boundary

value problems. Each of the one dimensional problems can be solved efficiently

by a standard computational scheme such as the Thomas algorithm.

In this article we construct and investigate block monotone iterative methods based on the Jacobi and Gauss–Seidel methods for solving coupled systems of nonlinear elliptic equations with quasimonotone nondecreasing reaction functions f

with α = 1, 2 .

In Section 2 we consider a nonlinear difference scheme which approximates the nonlinear elliptic problem (1) and describe the construction of the block monotone Jacobi and Gauss–Seidel iterative methods. A convergence analysis of the block monotone Jacobi and Gauss–Seidel iterative methods is discussed.

The theorem on uniqueness of a solution to the nonlinear difference scheme is proved. Section 3 presents numerical experiments.

2 Block monotone iterative methods

On ¯ ω = ω∪∂ω we introduce a rectangular mesh ¯ ω

= ¯ ω

× ¯ ω

= ω

∪∂ω

where ∂ω

is the boundary of the mesh ω

and

¯

ω

= {x

, i = 0, 1, . . . , N

; x

= 0 , x

= 1 ; h

= x

− x

} ,

¯

ω

= {y

, j = 0, 1, . . . , N

; y

= 0 , y

= 1 ; h

= y

− y

} . For a mesh function U(p

) = U

(p

), U

(p

)
with p

= (x

, y

) ∈ ¯ ω

we use the difference scheme

L

U

(p

) + f

(p

, U) = 0 , p

∈ ω

, α = 1, 2 , (2) U(p

) = g(p

) , p

∈ ∂ω

,

The linear difference operators L

are defined by

L

U

(p

) = −ε

D

U

(p

) + D

U

(p

)
,

where D

U

(p

) and D

U

(p

) for α = 1, 2 are the central difference ap- proximations to the second derivatives:

D

U

(p

) = U

− 2U

+ U

h

,

D

U

(p

) = U

− 2U

+ U

h

, U

≡ U

(p

) .

The vector mesh functions e U and b U are ordered upper and lower solutions, respectively, of (2) which satisfy the inequalities

U e

(p

) > b U

(p

) , p

∈ ¯ ω

, (3) L

U b

(p

) + f

(p

, b U) 6 0 6 L

U e

(p

) + f

(p

, e U) , p

∈ ω

, U b

(p

) 6 g

(p

) 6 e U

(p

) , p

∈ ∂ω

, α = 1, 2 .

For a given pair of ordered upper and lower solutions e U and b U we define the sector

h b U, e Ui =

U(p

) : b U

(p

) 6 U

(p

) 6 e U

(p

) , p

∈ ¯ ω

, α = 1, 2 

. We assume that on h b U, e Ui the vector function f(p

, U) in (2) satisfies the constraints

f

(p

, U)

uα

6 c

(p

) , U ∈ h b U, e Ui , α = 1, 2 , (4)

− f

(p

, U)

u_α0

> 0 , U ∈ h b U , e Ui , α

6= α , α, α

= 1, 2 , (5) for p

∈ ¯ ω

and where (f

)

≡ ∂f

/∂u

, (f

)

≡ ∂f

/∂u

and c

are non-negative bounded functions on ¯ ω

. The vector function f(p

, U) is quasimonotone nondecreasing on h b U, e Ui if it satisfies (5).

To construct block iterative methods we write the difference scheme (2) at an interior mesh point p

∈ ω

in the form

d

U

− l

U

− r

U

− b

U

− t

U

=

− f

(p

, U

, U

) + G

, (6)

where G

, like the boundary function g

, describes the boundary mesh points, and

l

= r

= ε

h

, b

= t

= ε

h

, d

= l

+ r

+ b

+ t

, α = 1, 2 , Define column vectors and diagonal matrices by

U

= (U

, . . . , U

)

, G

= (G

, . . . , G

y−1

)

,

F

(U

, U

) = f

(U

, U

), . . . , f

(U

, U

)

, L

= diag(l

, . . . , l

) , R

= diag(r

, . . . , r

) , for i = 0, 1, . . . , N

and where

F

(U

, U

) =

 F

(U

, U

) , α = 1 ,

F

(U

, U

) , α = 2 , α

6= α , (7) with symmetry F

(U

, U

) = F

(U

, U

) . Thus, L

U

is on the boundary and in G

, and R

U

is on the boundary and in G

x

. Then the difference scheme (2) is written in the form

A

U

− (L

U

+ R

U

) = −F

(U

, U

) + G

, (8) U

= (U

, U

) , i = 1, 2, . . . , N

− 1 , α = 1, 2 ,

where A

is the tridiagonal matrix with elements d

, l

and r

with j = 0, 1, . . . , N

. The elements of the matrices L

and R

are the coupling coefficients of a mesh point to U

and U

with j = 1, 2, . . . , N

− 1 . The upper { ˜U

} and lower { ^U

} sequences of solutions with number of iterations n > 1 are calculated by the following block Jacobi (η = 0) and Gauss–Seidel (η = 1) iterative methods:

A

Z

− ηL

Z

+ C

Z

= − K

(U

, U

) ,

K

(U

, U

) = A

U

− L

U

− R

U

+ F

(U

, U

) − G

,

where α = 1, 2 and i = 1, 2, . . . , N

− 1 , Z

=

 g

− U

, n = 1 ,

0 , n > 2 , i = 0, N

, (9)

Z

= U

− U

, η = 0, 1,

where U

= (U

, U

) , K

(U

, U

) are the residuals of the difference equations (8) on U

, and 0 is the zero column vector with N

− 1 components. The matrices C

are the diagonal matrices diag(c

, . . . , c

) where the c

= c

(p

) are defined in (4).

The mean-value theorem for vector-valued functions is F

(U

, U

) − F

(V

, U

) = F

(Y

, U

)

uα

[U

− V

] , (10) F

(U

, U

) − F

(U

, V

) = F

(U

, Y

)

u_α0

[U

− V

] , where the Y

lie between U

and V

, and the Y

lie between U

and V

, for i = 1, 2, . . . , N

− 1 , α

6= α , α, α

= 1, 2 . The partial derivatives are the diagonal matrices

(F

)

= diag (f

)

, . . . , (f

)

, (F

)

α0

= diag (f

)

α0

, . . . , (f

)

α0

,

where (f

)

and (f

)

, j = 1, 2, . . . , N

−1 , are calculated at Y

and Y

, respectively.

Theorem 1. Assume that f

with α = 1, 2 satisfies (4) and (5). Let ˜ U = ( ˜ U

, ˜ U

) and ^ U = ( ^ U

, ^ U

) be ordered upper and lower solutions of (2). Then for i = 0, 1, . . . , N

the upper sequence { ˜U

} generated by ( 9) with ˜ U

= ˜ U converges monotonically from above to a maximal solution ˜ V, and similarly, the lower sequence { ^U

} generated by ( 9) with ^ U

= ^ U converges from below to a minimal solution ^ V, such that,

U ^

6 ^ U

6 ^ V

6 ˜ V

6 ˜ U

6 ˜ U

, (11)

where the inequalities between vectors are in a component-wise sense, for

example, U

6 V

implies U

6 V

for all j = 0, . . . , N

.

Proof: Since ˜ U

is an initial upper solution (3), from (9) we have A

Z ˜

− L

Z ˜

+ C

Z ˜

= − K

( ˜ U

, ˜ U

) , i = 1, 2, . . . , N

− 1 ,

(12) Z ˜

6 0 , i = 0, N

, α = 1, 2 .

Since L

> O and (A

+ C

)

> O (Corollary 3.20, [ 6]) where O is the (N

− 1) × (N

− 1) null matrix, for i = 1 in (12) and ˜ Z

6 0 , we conclude that ˜ Z

6 0 . For i = 2 in ( 12), using L

> O and ˜ Z

6 0 , we obtain Z ˜

6 0 . Thus, by induction on i we prove that

Z ˜

6 0 , i = 0, 1, . . . , N

, α = 1, 2 . (13) Similarly, we can prove that

Z ^

> 0 , i = 0, 1, . . . , N

, α = 1, 2 . (14) We now prove that

U ^

6 ˜ U

, i = 0, 1, . . . , N

, α = 1, 2 . (15) Defining W

= ˜ U

− ^ U

for i = 0, 1, . . . , N

and α = 1, 2 , from (9) with i = 1, 2, . . . , N

− 1 and α = 1 we have

A

W

− L

W

+ C

W

= C

W

+ R

W

(16)

− h

F

( ˜ U

, ˜ U

) − F

( ^ U

, ˜ U

) i

− h

F

( ^ U

, ˜ U

) − F

( ^ U

, ^ U

) i

, and for i = 0, N

we have W

= 0. By the mean-value theorem (10), for i = 0, 1, . . . , N

we have

F

( ˜ U

, ˜ U

) − F

( ^ U

, ˜ U

) = F

(Q

, ˜ U

)

u1

h ˜ U

− ^ U

i

, F

( ^ U

, ˜ U

) − F

( ^ U

, ^ U

) = F

( ^ U

), Q

u2

h ˜ U

− ^ U

i

,

where ^ U

6 Q

6 ˜ U

for α = 1, 2 , and we conclude that (F

)

and (F

)

satisfy (4) and (5). Now with (16) we have, for i = 1, 2, . . . , N

− 1 ,

A

W

− L

W

+ C

W

= (C

− (F

)

) W

(17)

− (F

)

W

+ R

W

, and W

= 0 for i = 0, N

. Now with (4) and (5), and since W

> 0 for i = 0, 1, . . . , N

and α = 1, 2 , and R

> O , we obtain

A

W

+ C

W

> L

W

, i = 1, 2, . . . , N

− 1 , (18) W

= 0 , i = 0, N

.

Since (A

+ C

)

> O for i = 1, 2, . . . , N

− 1 , with i = 1 in (18) and W

= 0 , we conclude that W

> 0 . For i = 2 in ( 18), and using L

> 0 and W

> 0 , we obtain W

> 0 . Thus, by induction on i we prove that

W

> 0 , i = 0, 1, . . . , N

.

By following a similar argument we can prove (15) for α = 2 .

We now prove that ˜ U

and ^ U

with i = 0, 1, . . . , N

and α = 1, 2 are upper and lower solutions to (9), respectively. From (9) with α = 1 and using the mean-value theorem (10), we conclude that for i = 1, 2, . . . , N

− 1 ,

K

( ˜ U

, ˜ U

) = − C

− ∂F

(˜ E

, ˜ U

)

∂u

!

Z ˜

+ ∂F

( ˜ U

, ˜ E

)

∂u

Z ˜

− R

Z ˜

, (19)

where

U ˜

6 ˜ E

6 ˜ U

, i = 0, 1, . . . , N

, α = 1, 2 .

From (13), (14) and (15) we conclude that ∂F

/∂u

and ∂F

/∂u

satisfy (4) and (5). From (4), (5), (13) and since R

> O we conclude that

K

( ˜ U

, ˜ U

) > 0 , i = 1, 2, . . . , N

− 1 . (20)

Similarly,

K

( ˜ U

, ˜ U

) > 0 , i = 1, 2, . . . , N

− 1 . (21) From (3), (20) and (21) we conclude that ( ˜ U

, ˜ U

) for i = 0, 1, . . . , N

is an upper solution to (2). In a similar manner we obtain

K

( ^ U

, ^ U

) 6 0 , K

( ^ U

, ^ U

) 6 0 , i = 1, 2, . . . , N

− 1 , which means ( ^ U

, ^ U

) for i = 0, 1, . . . , N

is a lower solution to (2). By induction on n we can prove that { ˜U

} and { ^U

} with i = 0, 1, . . . , N

and α = 1, 2 are, respectively, monotone decreasing upper and monotone increasing lower sequences of solutions.

Now we prove that the limiting functions of the upper sequence { ˜U

} and lower sequence { ^U

} with i = 0, 1, . . . , N

and α = 1, 2 are, respec- tively, maximal and minimal solutions of (2). From (11) we conclude that lim

U ˜

= ˜ U

exists and

n

lim

Z ˜

= 0 , i = 0, 1, . . . , N

, α = 1, 2 . (22) Similar to (19), we have

K

( ˜ U

, ˜ U

) = − C

− ∂F

(˜ E

, ˜ U

)

∂u

!

Z ˜

− R

Z ˜

(23)

+ ∂F

( ˜ U

, ˜ E

)

∂u

Z ˜

, i = 1, 2, . . . , N

− 1 , where

U ˜

6 ˜ E

6 ˜ U

, i = 0, 1, . . . , N

, α = 1, 2 .

By taking the limit of both sides of (23), and using (13), it follows that

K

( ˜ U

, ˜ U

) = 0 , i = 1, 2, . . . , N

− 1 . (24)

Similarly, we obtain

K

( ˜ U

, ˜ U

) = 0 , i = 1, 2, . . . , N

− 1 . (25) From (24) and (25) we conclude that ( ˜ U

, ˜ U

) with i = 0, 1, . . . , N

is a maximal solution to the nonlinear difference scheme (2). In a similar manner, we can prove that

K

( ^ U

, ^ U

) = 0 , K

( ^ U

, ^ U

) = 0 , i = 1, 2, . . . , N

− 1 , which means that ( ^ U

, ^ U

) with i = 0, 1, . . . , N

is a minimal solution to

the nonlinear difference scheme (2). ♠

2.1 Convergent analysis

Assume that the reaction functions f

with α = 1, 2 satisfy the assumptions 0 < ^ c

(x, y) 6 f

(x, y, u)

uα

6 ˜c

(x, y) , (26)

0 6 − f

(x, y, u)

u_α0

6 q

(x, y) , α

6= α , α, α

= 1, 2 , (27) ρ = min

α=1,2

 min

(x,y)∈ ˜ω

^ c

(x, y)

> 0 , (28)

0 < β = max

α=1,2

 max

(x,y)∈ ˜ω

 q

(x, y)

^ c

(x, y)



< 1 , α

6= α , α, α

= 1, 2 . (29)

A stopping test for the block monotone iterative methods (9) is chosen to be max

α=1,2

k K

(U

)k

6 δ , k K

(U

)k

= max

16i6Nx−1

|K

(U

) | , (30) where δ is a prescribed accuracy.

The linear version of problem (2) is

L

W

(p

) + c

(p

)W

(p

) = Φ

(p

) , p

∈ ω

, (31)

W(p

) = g(p

) , p

∈ ∂ω

, α = 1, 2 ,

where W = (W

, W

) and the c

with α = 1, 2 are positive bounded functions.

We give an estimate of the solution to (31) in the following lemma.

Lemma 2. The solution to (31) satisfies

kW

k

6 max {kg

k

, kΦ

/c

k

} , α = 1, 2 , (32) where

kg

k

= max

pij∈∂ω^h

|g

(p

) | ,

Φ

c

= max

pij∈ω^h

Φ

(p

) c

(p

)

    . Samarskii [5] proves this lemma.

Theorem 3. Let assumptions (26)–(29) be satisfied. Then for the se- quence {U

} generated by the block monotone iterative methods ( 9) we have

kU

− U

k

6 1

(1 − β)ρ δ , (33)

where U

is a solution of the nonlinear difference scheme (2) and n

is the minimal number of iterations subject to (30).

Proof: The existence of a solution U

to the nonlinear difference scheme (2) is established in Theorem 1. From (2), for U

and U

, we have

L

U

(p

) + f

(p

, U

) = K

U

, U

, p

∈ ω

, U

(p

) = g

(p

) , p

∈ ∂ω

, α = 1, 2 ,

L

U

(p

) + f

(p

, U

) = 0 , p

∈ ω

, U

(p

) = g

(p

) , p

∈ ∂ω

, α = 1, 2 .

Letting W

= U

− U

for α = 1, 2 and using the mean-value theorem, we obtain

L

W

(p

) + f

(p

, H

)
+ u

W

(p

) =

− f

(p

, H

)

u_α0

W

(p

) + K

(U

, U

) , p

∈ ω

,

W

(p

) = 0 , p

∈ ∂ω

, α

6= α , α, α

= 1, 2 ,

where H

lies between U

and U

for α = 1, 2 . Using the maximum principle (32) we conclude that

kW

k

6 kK

(U

) 

f

(H

)

uα



k

+ k f

(H

)

u_α0

/ f

(H

)

uα

k

kW

k

. Letting W

= max

k W

k

and with (28) and (29) we obtain

W

6 (max

α=1,2

k K

(U

)k)ρ

+ β W

.

Now with (30) we have (33). Thus, we prove the theorem. ♠

2.2 Uniqueness of a solution

In this section we prove uniqueness of a solution of the discrete problem (2).

Theorem 4. Let assumptions (26)–(29) be satisfied. Then the nonlinear difference scheme (2) has a unique solution.

Proof: To prove the uniqueness of a solution to the nonlinear difference scheme (2), because of (11), it suffices to prove that ^ V

= ˜ V

, where ^ V

and ˜ V

are the minimal and maximal solutions. Substituting W

= ˜ V

− ^ V

into (2) we have

L

W

(p

) + f

(p

, ˜ V) − f

(p

, ^ V) = 0 , p

∈ ω

, W

(p

) = 0 , p

∈ ∂ω

, α = 1, 2 .

Using the mean-value theorem we obtain

L

+ f

(p

, Q

)

uα



W

(p

) = − f

(p

, Q

)

u_α0

W

(p

) ,

p

∈ ω

, W

(p

) = 0 , p

∈ ∂ω

, α

6= α , α, α

= 1, 2 ,

where ^ V

(p

) 6 Q

(p

) 6 ˜ V

(p

for α = 1, 2 . Using the maximum princi- ple (32) we conclude that

kW

k

6 k(f

(Q

))

α0

W

[(f

(Q

))

]

k

6 k(f

(Q

))

[(f

(Q

))

]

k

kW

k

. Using (29) we obtain

kW

k

6 βkW

k

. Let W = max

kW

k

so that

W(1 − β) 6 0 .

From (28) and since W > 0 we conclude that W = 0 . Thus, we prove the

theorem. ♠

As follows from Theorems 1 and 4, under assumptions (26)–(29), the sequences of solutions generated by the block Jacobi and Gauss–Seidel methods converge to the unique solution of the nonlinear difference scheme (2).

3 Numerical experiments

As a test problem we consider the gas-liquid interaction model [3] where reaction functions are

f

(u

, u

) = −σ

(1 − u

)u

, f

(u

, u

) = σ

(1 − u

)u

, (34) where u

> 0 and u

> 0 are concentrations of the gas and liquid, respectively, and σ

= const > 0 with α = 1, 2 are reaction rates.

We choose ε

= 1 , ε

= 0.1 , the boundary conditions g

(x, y) = 0 and g

(x, y) = 1 , (x, y) ∈ ∂ω in (1), and σ

= 1 for α = 1, 2 . The pairs ( ˜ U

, ˜ U

) = (1, 1) and ( ^ U

, ^ U

) = (0, 0) are ordered upper and lower solutions.

From (34) we conclude that

(f

)

= u

6 1 , −(f

)

= 1 − u

> 0 ,

(f

)

= 1 − u

6 1 , −(f

)

= u

> 0 .

Table 1: Numerical error and order of convergence of the nonlinear scheme (2).

N 8 16 32 64 128

E 0.0071 0.0017 4.47 × 10

1.06 × 10

2.13 × 10

γ 1.97 2.01 2.06 2.32

Table 2: Number of iterations and CPU time for the block methods.

N 8 16 32 64 128

block Jacobi method

# of iterations 101 397 1577 6299 25189 CPU (s) 0.02 0.11 0.91 14.17 225.99

block Gauss–Seidel method

# of iterations 51 180 762 3084 12370 CPU (s) 0.01 0.06 0.47 7.34 117.62

It follows that f

with α = 1, 2 satisfy (4) with c

= 1 and (5). Since the exact solution of the test problem is unavailable, we define the numerical error and the order of convergence of the numerical solution, respectively, as

E(N) = max

α=1,2

 max

pij∈ ¯ωh

 U

(p

) − U

(p

)  



, γ(N) = log

 E(N) E(2N)



,

where U

(p

) with α = 1, 2 are the approximate solutions generated by (9),

n

is the minimal number of iterations subject to (30), and U

(p

) with

α = 1, 2 are reference solutions with number of mesh points N = 512 .

Table 1 presents the error E(N) and order of convergence γ(N) for different

values of N

= N

= N . This table indicates that the numerical solution of

the nonlinear difference scheme (2) converges to the reference solution with

second-order accuracy. The numerical and reference solutions are calculated by

the block Jacobi or Gauss–Seidel methods. Tables 2 and 3 show that the block

Gauss–Seidel method converges faster than the block Jacobi method, and the

block monotone methods (Table 2) converge faster than the corresponding

monotone Gauss–Seidel and Jacobi methods (Table 3).

Table 3: Number of iterations and CPU time for the Jacobi and Gauss–Seidel methods.

N 8 16 32 64 128

Jacobi method

# of iterations 190 771 3092 12378 49520 CPU (s) 0.08 0.11 1.09 16.15 261.28

Gauss–Seidel method

# of iterations 97 388 1548 6191 24762 CPU (s) 0.12 0.40 0.53 8.58 141.37

References

[1] Boglaev, I., A block monotone domain decomposition algorithm for a semilinear convection-diffusion problem, J. Comput. Appl. Math., 173(2005), 259–277. doi:10.1016/j.cam.2004.03.011 C80

[2] Boglaev, I., Monotone iterates for solving systems of semilinear elliptic equations and applications, ANZIAM J, Proceedings of the 8th Biennial Engineering Mathematics and Applications Conference, EMAC-2007, 49(2008), C591–C608. doi:10.21914/anziamj.v49i0.311 C80

[3] Pao, C. V., Nonlinear parabolic and elliptic equations, Springer-Verlag (1992). doi:10.1007/978-1-4615-3034-3 C80, C91

[4] Pao, C. V., Block monotone iterative methods for numerical solutions of nonlinear elliptic equations, Numer. Math., 72(1995), 239–262.

doi:10.1007/s002110050168 C80

[5] Samarskii, A., The theory of difference schemes, CRC Press (2001).

https://www.crcpress.com/The-Theory-of-Difference-Schemes/

Samarskii/p/book/9780824704681 C89

[6] Varga, R. S., Matrix iterative analysis, Springer-Verlag (2000).

doi:10.1007/978-3-642-05156-2 C85

Author addresses

1. M. Al-Sultani, School of Fundamental Sciences, Massey University, Palmerston North, New Zealand; Faculty of Education for Pure Sciences, University of Kerbala, Kerbala, Iraq.

mailto:[email protected]

2. I. Boglaev, School of Fundamental Sciences, Massey University, Palmerston North, New Zealand.

mailto:[email protected]