SOLVING FULLY PARAMETERIZED SINGULARLY
PERTURBED NON-LINEAR PARABOLIC & ELLIPTIC PDE’S
BY EXPLICIT APPROXIMATE INVERSE
FE MATRIX ALGORITHMIC METHODS
G.A. GRAVVANIS1, K.M. GIANNOUTAKIS1 and E.A. LIPITAKIS2
1
Department of Electrical and Computer Engineering, School of Engineering,
Democritus University of Thrace, 12, Vas. Sofias street, GR 671 00 Xanthi, Greece;
email: {ggravvan, kgiannou}@ee.duth.gr
2
Department of Informatics, Athens University of Economics and Business,
76, Patision street, GR 104 34 Athens, Greece; Email: eal@aueb.gr
Abstract: A class of generalized
approximate inverse finite element
matrix algorithmic methods for solving
non-linear parabolic and elliptic PDE’s,
is presented. Fully parameterized
singularly perturbed non-linear parabolic
and elliptic PDE’s are considered and
explicit
preconditioned
generalized
conjugate gradient - type schemes are
presented for the efficient solution of the
resulting non-linear systems of algebraic
equations. Applications of the proposed
algorithmic methods on characteristic
two-dimensional non-linear boundary
value and initial value problems are
discussed and numerical results are
given.
Keywords: finite element method, nonlinear
systems,
preconditioning,
preconditioned
conjugate
gradient
method, parallel iterative methods.
CR categories: F.2.1, G.1.0, G.1.3,
G.1.8, G.4
AMS (MOS): 65F10, 65F50, 65M60,
65N30, 65Y05
1. INTRODUCTION
In the last decades research has been
directed in the study of a class of
boundary value problems and the
behavior of the approximate solutions of
the resulting linear systems by
considering a small positive perturbation
parameter, affecting the derivative of
highest order, cf. [8,11,14,15,18].
Following this approach, in this article
we consider a class of generalized fully
parameterized singularly perturbed (sp)
non-linear initial and boundary value
problems and we study the way that the
sp parameters variation affects their
numerical solution.
Let us consider the following fully
parameterized
non-linear
parabolic
P.D.E. in two space variables:
u
βu
ε
- ε Δu  ε  u  u  αe
t t 2
1 x
y
ε , ε 2 , ε  0  (x,y)Ω, t0, (1)
t
1
with initial conditions:
u(x,y,0)=gi(x,y), 0  x,y  1,
(1.a)
and boundary conditions:
u(x,y,t) = gb (x,y), t0,
(1.b)
where ε , ε 2 and ε
are singular
t
1
perturbation parameters affecting the
derivative with respect to time and the
derivatives with respect to the space
variables
respectively.
The
real
parameters α and β affect the non-linear
term, while certain values of these
parameters lead to highly non-linear
terms.
Then by considering the upwind
(downwind) stable discretization scheme
of accuracy O(h) for given ε1 (assuming
a uniform mesh-size h is used), viz.,
 u(x  h, y)  u(x, y)
, ε 0
 ε1
1
h
(2)
ε ux  
1
u(x, y)  u(x  h, y)
ε
, ε 0
1
1
h
t

1
t
and u t  u i, j  u i, j /Δt .
The (weakly) diagonal dominant linear
system derived from the finite element
method of the partial differential
equation, cf. (1)-(1.a)-(1.b), is
Au=s,
(3)
where A is a non-singular sparse positive
definite (nn) matrix, with all the offcenter band terms grouped into a regular
band of width  , of the structure as given
in (4).
Note that the elements bi, ai, ci, vη,θ and
tκ,λ of the coefficient matrix A are
functions of the considered sp parameters
ε , ε 2 , ε and the real parameters α, β.
t
1
An important achievement over the last
two decades for solving sparse finite
element linear systems is the appearance
and efficient use of Preconditioned


methods, cf. [1,2,3,4,6,7,9,10,12,13,17].
The resulting preconditioned form of the
non-linear system (3) is
M A u = M s,
(5)
where M is a suitable preconditioner.
Many researchers have presented several
forms of the preconditioner M. In recent
years, Sparse Approximate Inverse
Preconditioning has been introduced,
based on factorized sparse approximate
inverses or minimization of some
convenient norm, cf. [1,2,7,9,11,16], and
on
approximate
inversion
of
corresponding incomplete factors, cf.
[2,9,13,16]. Furthermore, Approximate
Inverse Matrix techniques (AIM) have
been proposed and have been efficiently
used in conjunction with explicit
preconditioned
conjugate
gradient
schemes, which are suitable for solving
linear systems on multiprocessor and
multi-computer systems, cf. [4,5,10,13].
Finally,
the
performance
and
applicability
of
the
explicit
preconditioned generalized conjugate
gradient type schemes is illustrated by
solving singular perturbed elliptic and
parabolic non-linear problems and
numerical results are given.
2. FINITE ELEMENT APPROXIMATE INVERSES
In this section we present explicit
generalized approximate inverse finite
element matrix algorithmic techniques by
computing the elements of a class of
inverses, cf. [6,10,13].
Let us now assume the generalized
approximate LU-type factorization of the
coefficient matrix A, cf. [9], such that:
A  L r U r , r [1,..., m - 1),
(6)
where r is the “fill-in” parameter, i.e. the
number of outermost off-diagonal entries
at semi-bandwidth m, and L r , U r , viz.


  b1

a

2

m 

A 
































m
c
1
b
2


c
2






































Τ(tk,λ)
0
a
m-1
b
m-1
am
c
m-1
bm
0
cm
V(vη,θ)
an
  w1
 d

1


m 
c
n-1
bn

Lr  































w
2
d
2
0
0
w
m-1
d
m-1
0
E(hi,j)
d

































1
Ur 
wm
dm
 r 
m
g
1
1
g

wn
n-1
H(hi,j)
g
m-1
1





































(7)


2
1
(4)
gm
g



















 
 r
 










n -1
1
(8)
are sparse lower and upper (with unit
diagonal elements) triangular matrices
respectively of the same profile as the
coefficient matrix A, cf. (4).
Then, the elements of the decomposition
factors L r and U r were computed by the
FEALUFA-2D algorithm, cf. [6,9]. The
memory requirements of the FEALUFA-2D
algorithm are  O2r  4  6 n words. The
computational work required by the
factorization
process
is
multiplicative
 O r    12  3 n


operations, cf. [6,9].
Let M δl, δu  (μ ), i[1, n], j[max(1, ii, j
r
δl+1), min(n, i+δu)], be the generalized
approximate inverse of the coefficient
matrix A, i.e.


M δl, δu = L U  1.
r
r r
(9)
A class of approximate inverses can be
obtained by retaining δl and δu diagonal
vectors, cf. [6,10,13], by solving recursively
the following systems:
1
M δl, δu L =  U 
and
r
r  r
.
(10)
1
 
U r M δl, δu =  L 
r
 r
Then, the elements of the generalized
approximate inverse were computed by the
Optimized
Generalized
Approximate
Inverse
Finite
Element
Matrix
(OGAIFEM-2D) algorithm, cf. [6,13]. The
memory requirements of the OGAIFEM2D algorithm are  Oδl  δu  n words.
The computational work required by
generalized approximate inverse process is
 O(δl  δu)2r  2  1n  multiplicative
operations, cf. [6,13].
It should be noted that this class of
generalized approximate inverse includes
various families of approximate inverses
according to the requirements of accuracy,
storage and computational work, as can be
seen by the following diagrammatic
relation:
class I class II class III class IV
1
~ δl,δu
δl,δu
δl,δu
 M  Mr m1  Mr m1  Mr
 M
i
(11)
where M is the exact inverse resulting in a
direct method, i.e. r=m-1 and δl=n, δu=δl-1
with the disadvantage of high memory
requirements and computational work for
large order systems. The entries of the class
I inverse have been retained after the
computation of the exact inverse ( r =m-1,
δl=n, δu= δl-1) by retaining only δl and δu
elements in the lower and upper part of the
exact inverse. The entries of the class II
inverse have been computed and retained
during the computational procedure of the
(approximate) inverse ( r =m-1), while the
entries of the class III inverse have been
retained after the computation of the
approximate inverse (r  m-1), cf. [13]. The
class IV of the generalized approximate
inverse retains only the diagonal elements,
i.e. δl=1 hence the diagonal entries of the
sparse lower matrix L r , cf. (7), resulting in
a fast inverse algorithm.
It should be noted that if the widthparameter  = 1 then the algorithm reduces
to one for solving linear systems of semibandwidth m, by the ALUFA-2D and
OGAIM-2D
algorithm,
which
is
encountered usually in solving 2D boundary
value problems by the finite difference
method.
A
3. EXPLICIT PRECONDITIONED
CONJUGATE
GRADIENT
METHODS
In this section we present a class of explicit
preconditioned
generalized
conjugate
gradient schemes, based on the generalized
approximate inverses.
The Explicit Preconditioned Generalized
Conjugate Gradient Square (EPGCGS)
method can be expressed by the following
compact algorithmic scheme:
Let u be an arbitrary initial approximation
0
to the solution vector u. Then,
u 0  0 and e 0  0,
δl,δu (s  Au ),
solve r  M r
0
0
σ 0  r and p = σ , r .
set
0
0
0 0
set


(12)
(13)
(14)
Then, for i=0, 1, ..., (until convergence)
compute the vectors u
and
,r
,σ
i +1 i +1
the scalar quantities α , β
i
i +1
i+1
as follows:
q = Aσ ,
i
i
α = p  σ , M δl, δu q  ,
i i  0 r
i
calculate
compute e
= r + β e - α M δl,δu q ,
i +1 i i i i r
i
d = r +β e +e
i i
i i
i +1
u
u +α d ,
and
i +1
i
i i
q = Ad ,
form
i
i
δu q ,
compute r
= r - α M δl,
r
i
i +1
i
i
p
= σ ,r
compute
,
i +1
0 i +1
β =p
p
i +1 i +1 i
and σ  r + 2β e
+ β2 σ .
i+1 i + 1
i +1 i +1 i +1 i


(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Then the computational complexity of the
EPGCGS method requires O[(2δl+2δu
+4  +15)n mults + 8n adds]ν operations,
where ν is the number of iterations required
for the convergence to a certain level of
accuracy and  is the width parameter of
the coefficient matrix A at semi-bandwidth
m, cf. (4).
In the following we present the Explicit
Preconditioned Generalized BIconjugate
Conjugate Gradient-STAB (EPGBICGSTAB) method, which can be expressed by
the following compact scheme:
Let u be an arbitrary initial approximation
0
to the solution vector u. Then,
set
(25)
u0  0 ,
compute r  s  Au ,
(26)
0
0
set
r '  r , ρ = α = ω = 1 , (27)
0 0
0
0
and
(28)
v = p = 0.
0
0
Then, for i=0, 1, ..., (until convergence)
compute the vectors u , r and the scalar
i i
quantities α, β, ω as follows:
i
calculate ρ =  r ' , r  ,
(29)
i  0 i -1 
i
β= ρ

(30)
x = ri-1 - αv ,
i
i
δl,δu
z = Mr
xi
i
t  Az
i
i
(35)
compute
form
and
ρ


,
αω
i -1
i -1
compute p = ri-1 + β p  ω v
,
i
i -1
i -1 i -1
δl,δu
y = Mr
pi ,
form
i
and
v  Ay ,
i
i
calculate α = ρ  r ' , v  ,
i  0 i
and

(31)
(32)
(33)
(34)
(36)
(37)
set
δl, δu
δl, δu   δl, δu
δl, δu 
ω =  M
t ,M
x  M
t ,M
t 
i  r
i r
i  r
i r
i
(38)
+ αy + ω z i
compute u i  u
(39)
i -1
i
i
ri  x - ω t i .
and
(40)
i
i
Then the computational complexity of the
EPGBICG-STAB method requires 
O[(4δl+4δu +4  +16)n mults + 6n adds]ν
operations, where ν is the number of
iterations required for the convergence to a
certain level of accuracy and  is the width
parameter of the coefficient matrix A at
semi-bandwidth m, cf. (4).
The
effectiveness
of
the
explicit
preconditioned
generalized
conjugate
gradient-type methods is related to the fact
that the generalized approximate inverse
exhibits a similar “fuzzy” structure as the
original coefficient matrix A and is a close
approximant to the coefficient matrix A, cf.
[6,13].
4. NUMERICAL RESULTS
In this section we examine the applicability
and
effectiveness
of
the
explicit
preconditioned
generalized
conjugate
gradient schemes for solving characteristic
singular perturbed non-linear boundary
value and initial value problems.
Model Problem I: Let us consider the
following non-linear parabolic P.D.E. in two
space variables:
u
βu
ε
- ε Δu  ε  u  u  αe ,
t t 2
1 x
y
ε , ε 2 , ε  0  (x,y)Ω, t0,
t
1
with initial conditions:
(41)
u(x,y,0)=g(x,y), 0  x,y  1,
and non-linear boundary conditions:
(41.a)
u(x,y,t) =0,
t0.
(41.b)
A non-overlapping triangular network
resulting in a hexagonal mesh covered the
region, for the computation of an
approximate solution of the perturbation
problem. Then, by using backward
differences for u , the (downwind) stable
t
scheme of order O(h) for u , u and the
x y
finite element scheme, with a row-wise
ordering used such that the width parameter
 of the band was kept to low values, i.e.
  3 , then the resulting coefficient matrix
is of the form given in (4). The “fill-in”
parameter was set to r=2. The initial guess
was u(0)=1.0.
The termination criterion for the explicit
preconditioned
generalized
conjugate
gradient - type scheme was  r i  10-4,

where r i is the recursive residual. The
criterion for the termination of the Newton
and the steady state solution was
 (k  1)  u (k)  / 1  u (k  1)   10 - 4 ,

max  u j
j  
j

j
j  [1,n].
The quasi-linearized Newton scheme is of
the following form:
h 2  ε t
βu (k)  (k  1)
(k  1)
 αβe
u
L u
h
i,
j
i,
j


ε  Δt

2
hε
(k  1) (k  1) (k  1) 
, (42)
 1  2u
-u
u
i,
j
i - 1, j
i, j  1 
ε 
2
 (k)

h2  εtu
(k) βu (k) 
 
 (1  βu )αe

ε  Δt

2

with Lh denoting here the corresponding
descritized finite element operator.
Numerical results for the steady state
solution using backward differences (θ=1)
combined with Newton compact iterations
in conjunction with the EPGCGS and
EPGBICG-STAB method, based on the
OGAIFEM-2D algorithm, for several
values of the parameters ε , ε 2 , ε , α, β,
t
1
the time-step Δt and the “retention”
parameter δl, with θ=1, n=361, m=20, are
presented in Table 1 and 2 respectively.
Model Problem II: Let us consider the
following non-linear P.D.E. in two space
variables:
βu
ε Δu  ε  u  u  αe ,
2
1 x
y
(x,y)Ω,
ε 2 , ε  0
1
with boundary conditions:
(43)
u(x,y) =0,
(43.a)
A non-overlapping triangular network
resulting in a hexagonal mesh covered the
domain. The width parameter at semibandwidth m was chosen to be   3 and
the “fill-in” parameter was set to r=2. The
initial guess was u(0)=1.0.
The termination criterion for the normalized
preconditioned conjugate gradient - type
schemes was  r i  10-4, where r i is the

recursive residual. The criterion for the
termination of the Newton method was
 (k  1)  u (k) /1  u (k  1)   10 - 4 , j[1,n].

max  u j
j  
j

j
εt
ε2=α=β
ε1
1
0.01
0.000001
1.0
1
1
0.01
0.000001
1
0.01
0.000001
1
1
0.01
0.01
0.000001
0.01
0.000001
0.01
0.01
0.000001
1
0.01
0.000001
1
1
0.01
0.000001
0.01
0.000001
0.000001
0.01
0.01
0.000001
0.01
0.000001
0.000001
0.01
0.000001
Δt
0.05
0.01
0.05
0.01
0.05
0.01
0.05
0.01
0.05
0.01
0.05
0.01
no of Newton
number of EPGCGS
outer method
iterations
iter
(inner
δl=1 δl=2 δl=m δl=2m
s.s.s.
iter.)
4
12
28*
27
23
15*
4
12
28*
27
23
14
4
12
27
27
23
14
6
21
45
43
34
26
6
21
44
43
34
26
6
21
44
43
34
26
3
7
10
11
9**
7
3
7
10
11
9**
7
3
7
10
11
9**
7
3
8
12
14
11
9
3
8
12
14
11
9
3
8
12
14
11
9
4
12
28
27
23
15*
4
12
27
27
23
14
6
21
45
43
34
26
6
21
45
43
34
26
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
2
5
7
8
6
5
†
†
3
4
22
16
4
12
28*
27
23
15*
†
3
4
89*** 24
16
6
21
45
43
34
26
Table 1: The convergence behavior for steady state solution of the Newton method in
conjunction with the EPGCGS method for various values of ε , ε 2 , ε , α, β, δl,
t
1
δu=δl-1 and the time step Δt, with θ=1, n=361, m=20, r=2 and   3 .
* The number of inner iterations (Newton) was 13 iterations
** The number of inner iterations (Newton) was 9 iterations
*** The number of inner iterations (Newton) was 6 iterations
† Convergence criteria not met
εt
ε2=α=β
ε1
1
0.01
0.000001
1.0
1
1
0.01
0.000001
1
0.01
0.000001
1
1
0.01
0.01
0.000001
0.01
0.000001
0.01
0.01
0.000001
1
0.01
0.000001
1
1
0.01
0.000001
0.01
0.000001
0.000001
0.01
0.01
0.000001
0.01
0.000001
0.000001
0.01
0.000001
Δt
0.05
0.01
0.05
0.01
0.05
0.01
0.05
0.01
0.05
0.01
0.05
0.01
no of
outer
iter
s.s.s.
4
4
4
6
6
6
3
3
3
3
3
3
4
4
6
6
2
2
2
2
2
2
2
2
2
2
2
4
Newton
method
(inner
iter.)
12
12
12
21
21
21
7
7
7
8
8
8
12
12
21
21
5
5
5
5
5
5
5
5
5
5
2
12
number of EPGBICG-STAB
iterations
δl=1 δl=2
35
35
35
59
63
63
15
15
16
19
19
19
35
35
59
63
11
11
11
11
11
11
11
11
11
11
33
35
35
59
61
61
15
15
15
19
19
19
32
35
59
61
11
11
11
11
11
11
11
11
11
11
2
32
†
35
δl=m
δl=2m
30
30
30
49
50
50
15
15
15
18
18
18
30
30
49
50
11
11
11
11
11
11
11
11
11
11
2
30
26
26
26
45
45
45
14
14
14
16
16
16
26
26
45
45
10
10
10
10
10
10
10
10
10
10
†
26
†
5923 59
49
45
*CG
Table 2: The convergence behavior for steady state solution of the Newton method in
conjunction with the EPGBICG-STAB method *CG
for various values of
979
ε , ε 2 , ε , α, β, δl, δu=δl-1 and the time step Δt, with θ=1, n=361, m=20, r=2
t
1
(5,14)252
and   3 .
† Convergence criteria not met
6
21
ε2=α=β
0.01
0.000001
number of EPGCGS iterations
ε1
Newton method
(outer iter.)
δl=1
δl=2
δl=m
δl=2m
0.01
0.000001
0.000001
3
3
3
14
14
14
14
14
14
12
12
12
7
7
7
Table 3: The convergence behavior for the Newton method in conjunction with the
EPGCGS method for various values of ε 2 , ε , α, β, δl, δu=δl-1 with n=361,
1
m=20, r=2 and   3 .
ε2=α=β
1
0.01
0.000001
ε1
1
0.01
0.000001
0.01
0.000001
0.000001
Newton method
(outer iter.)
3
3
3
2
2
2
number of EPGBICG-STAB
iterations
δl=1
δl=2
δl=m
δl=2m
15
15
15
12
12
12
14
14
14
11
11
11
12
12
12
9
9
9
9
9
9
7
7
7
Table 4: The convergence behavior for the Newton method in conjunction with the
EPGBICG-STAB method for various values of ε 2 , ε , α, β, δl, δu=δl-1 with
1
n=361, m=20, r=2 and   3 .
The quasi-linearized Newton scheme is
of the following form:
h 2  βu (k)  (k  1)
(k  1)
αβe
u
L u


h
i,
j
i,
j
ε 

2
hε
(k  1) (k  1) (k  1) 
, (44)
 1  2u
-u
u
i - 1, j
i, j  1 
ε  i, j
2
h2 
(k) βu (k) 
  (1  βu )αe

ε 

2

with Lh denoting the descritized finite
element operator.
Numerical results for the Newton
compact iterations in conjunction with
the EPGCGS and EPGBICG-STAB
method based on the OGAIFEM-2D
algorithm, for several values of the
parameters ε 2 , ε , α, β, and the
1
“retention” parameter δl, with n=361,
m=20, are presented in Table 3 and 4
respectively.
It should be stated that the convergence
criteria for the steady state solution of the
Newton
compact
iterations
in
conjunction with the EPGCGS method
for ε2=α=β=1.0 and ε1=1, 0.01, 0.000001,
for various values of the “retention”
parameter δl, with n=361 and m=20 were
not met.
It should be mentioned that the explicit
approximate inverse preconditioning
schemes
have
been
efficiently
implemented on multiprocessor and
multi-computer systems, cf. [4,5].
Finally,
the
explicit
generalized
approximate inverse preconditioning
methods can be efficiently used for
solving three-dimensional highly nonlinear partial differential equations.
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