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Robust exponential convergence of
hp-FEM for singularly perturbed
systems of reaction-diffusion equations
Christos Xenophontos
Department of Mathematics and Statistics
University of Cyprus
joint work with
J. M. Melenk (TU Wien)
and
L. Oberbroeckling (Loyola Univ. MD)
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2
The Model Problem
Find ( ) ( ), ( ) such thatT
x u x v xU
11 12
21 22
( ) ( ) ( )( ) , ( ) are given.
( ) ( ) ( )
a x a x f xx x
a x a x g x
A F
, ( ) ( ) ( ) ( ) in (0,1)
(0) (1)
x x x x I
E U A U F
U U 0
where
2
,
2
0,0 1,
0
E and
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3
Assumptions:
2 T T x I ξ Aξ ξ ξ ξ
(1) The functions aij(x), f (x),g(x) are analytic, and
(2) The matrix A is pointwise positive definite, i.e. fixed
> 0, such that
( )
0,
( )
0,
( )
0,
! ,
! ,
! , , 1, 2
n n
f fI
n n
g gI
n n
ij a aI
f C n n
g C n n
a C n n i j
, , , , , ,f g f g a aC C C
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4
Remark:
Both components of the solution have boundary
layers at x = 0 and x = 1, of width O(ln), but the
second component has an additional sublayer of
width O(ln).
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4
Remark:
Both components of the solution have boundary
layers at x = 0 and x = 1, of width O(ln), but the
second component has an additional sublayer of
width O(ln).
This is illustrated in the figure below, which shows
the solution corresponding to
7 / 2 22 1 1
, ( ) , 10 , 101 2 1
A f x
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5
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6
Variational Formulation
2
1
0, : ( , )TB F u v H I U V V V
2
1
0Find such thatH I U
2 2
11 12 21 22
, , ,
+ , ,
B u u v v
a u a v u a u a v v
U V
where, with the usual L2(Ι) inner product,
, ,F f u g v V
,
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7
It follows that the bilinear form is coercive, i.e.
22 1
0, EB H I U U U U
2 2 2 2 22 2 21, 1, 0, 0,E I I I Iu v u v Uwhere
denotes the energy norm. We also have the a-priori
estimate
2 2,
0, 0,max 1,
I
E I I
Af g
U
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8
Scale separation:
The relationship between and determines the number
and nature of the layers. Correspondingly, there are four
cases:
(I) The “no scale separation case” which occurs when
neither μ/1 nor ε/μ is small.
(II) The “3-scale case” in which all scales are separated
and occurs when μ/1 is small and ε/μ is small.
(III) The first “2-scale case” which occurs when μ/1 is not
small but ε/μ is small.
(IV) The second “2-scale case” which occurs when μ/1 is
small but ε/μ is not small.
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Theorem 1:
There exist constants C, b, δ, q, γ > 0 independent of ε
and
μ, such that the following assertions are true for the solution
U:
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Theorem 1:
There exist constants C, b, δ, q, γ > 0 independent of ε
and
μ, such that the following assertions are true for the solution
U:
( ) 1/2 1,
ma( x ,I)n
n n
IC n
U
9
-
Theorem 1:
There exist constants C, b, δ, q, γ > 0 independent of ε
and
μ, such that the following assertions are true for the solution
U:
( ) 1/2 1,
ma( x ,I)n
n n
IC n
U
( ) / /
, ,,
( , )/( )
( , )/( )
ˆ where
,
(II)
( )
ˆ ( )
BL BL
n n n b b
I E II
n dist x In n
BL
n dist x In n
BL
C n C e e
x C e
x C e
U W U U R
W R R
U
U
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Theorem 1:
There exist constants C, b, δ, q, γ > 0 independent of ε
and
μ, such that the following assertions are true for the solution
U:
( ) 1/2 1,
ma( x ,I)n
n n
IC n
U
( ) / /
, ,,
( , )/( )
( , )/( )
ˆ where
,
(II)
( )
ˆ ( )
BL BL
n n n b b
I E II
n dist x In n
BL
n dist x In n
BL
C n C e e
x C e
x C e
U W U U R
W R R
U
U
Additionally, the second component satisfies the
sharper estimate
ˆˆ of BLv U
9
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10
2
( , )/( )ˆ ( )n dist x In nv x C e
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10
2
( , )/( )ˆ ( )n dist x In nv x C e
/
, ,
( ) 1
,
( , )/( )
(III) ˆ If / , where
max ,
ˆ ( )
BL
b
I E I
nn n
I
n dist x In n
BL
q
Ce
C n
x C e
U W U R
R R
W
U
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Additionally, the second component
satisfies the sharper estimate
ˆˆ of BLv U
2
( , )/( )ˆ ( )n dist x In nv x C e
/
, ,
( ) 1
,
( , )/( )
(III) ˆ If / , where
max ,
ˆ ( )
BL
b
I E I
nn n
I
n dist x In n
BL
q
Ce
C n
x C e
U W U R
R R
W
U
2
( , )/( )ˆ ( )n dist x In nv x C e
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11
2 /
, ,
( )
,
( , )/( )
where
( / )
(IV)
( ) / /
BL
b
I E I
n n n
I
n n dist x In
BL
C e
C n
x C e
U W U R
R R
W
U
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11
2 /
, ,
( )
,
( , )/( )
where
( / )
(IV)
( ) / /
BL
b
I E I
n n n
I
n n dist x In
BL
C e
C n
x C e
U W U R
R R
W
U
Remark:
The approximation of U will be constructed according to
the above regularity results, i.e. the mesh and polynomial
degree distribution will be chosen appropriately, based on
the relationship between ε and μ .
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12
Discretization
As usual, we seek 22 1
0 s. t.N NV H I U
and we have
2
,N NB F V U V V V
2
N NE EV U U V U V
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Discretization
As usual, we seek 22 1
0 s. t.N NV H I U
and we have
2
,N NB F V U V V V
2
N NE EV U U V U V
The space VN is defined as follows:
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Discretization
As usual, we seek 22 1
0 s. t.N NV H I U
and we have
2
,N NB F V U V V V
2
N NE EV U U V U V
The space VN is defined as follows:
Let Pn(t) denote the nth Legendre polynomial and define
1 21 1
( ) 1 , ( ) 12 2
2 1 31
2 3 1( ) ( ) ( ) ( ) , 3,..., 1
2 2(2 3)i i i i
iP t dt P P i p
i
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Then with Πp the space of polynomials of degree p
over [–1, 1], we have
1 2 3 1, , , ,p pspan
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Then with Πp the space of polynomials of degree p
over [–1, 1], we have
1 2 3 1, , , ,p pspan Now, partition the domain I = (0, 1)
by
and set
0 10 1Mx x x
1 1, , , 1,...,j j j j j jI x x h x x j M
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Then with Πp the space of polynomials of degree p
over [–1, 1], we have
1 2 3 1, , , ,p pspan Now, partition the domain I = (0, 1)
by
and set
0 10 1Mx x x
1 1, , , 1,...,j j j j j jI x x h x x j M
Also define the standard (or master) element
IST = (–1, 1), and note that it can be mapped onto
the jth element by the linear mapping
13
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14
11 1
( ) 1 12 2
j j jx Q x x
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14
11 1
( ) 1 12 2
j j jx Q x x
The finite element space VN is then defined as
10, : ( ) , 1, ,jN j pV u H I u Q j M p
where 1 2, , , Mp p pp
degrees assigned to the elements. We have
1
dim , 1M
N i
i
V p
p
is the vector of polynomial
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15
An hp finite element method – error
estimates
Definition: Spectral Boundary Layer Mesh
0, and 0 1p
2
1
0, : , withNS p V p H
define
0,1 , 1/ 2
0, , ,1 ,1 ,1 , 1/ 2
0, ,1 ,1 , 1/ 2
p
p p p p p
p p p p
For
The polynomial degree is taken to be uniformly p over all
elements.
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[0,1], p
►If both ε and μ are large (and no layers are present), then
0, ,1 ,1 ,p p p
In practice, the mesh is constructed as follows:
►If both ε and μ are small (i.e. 0 < ε < μ
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where U is the exact solution and UN is the finite element
solution computed using the Spectral Boundary Layer Mesh.
p
N ECe U U
Theorem 2: There exist constants C, , > 0, depending
only on the input data, such that
Sketch of Proof:
We utilize the previously stated regularity results, along
with
the following approximation results, the first one being
used
for the approximation of the smooth part(s) and the boundary
layers (within the layer).
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Lemma 1: Let Ij be an interval of length hj and let V C(IST)
satisfy for some Cu, γu > 0, K 1,
( ),
max , 1,2,3...j
nn n
u uIC n K n
V
Then there exist η, β, C > 0, depending only on γu such
that, under the condition ,j
j
h K
p we have
1/2
1
0,0,
,jj j
j
j
pj
j p p uI
jI
h Kh CC e
p
V V V V
where : H1(Ij) → Πp is a linear operator that satisfies
( ) ( ).jp j j
I I V V
jp
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Lemma 2: Let ν > 0 and let u satisfy
( , )/( ) ( ) .dist x Iuu x u x C e x I
/1 1 0,( ,1 )0,( ,1 )
,uu u u u CC e
Let Δ be an arbitrary mesh on I with mesh points ξ and 1 – ξ
where ξ (0, ½). Then the piecewise linear interpolant π1u
Satisfies on (ξ, 1 – ξ ):
for some C > 0 independent of ν.
The next result is used for the approximation of the
boundary
layers outside the layer.
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The proof is separated in four cases corresponding to the
four
cases stated in the regularity results. In Case I
(asymptotic
case), Theorem 1 and Lemma 1 give the desired result. In
Case II (3 scale separation), Theorem 1 and Lemma 1 allow us
to handle the smooth part and the remainder in the
expansion.
For the layers, we use Lemma 2 for their approximation
outside the layer region and Lemma 1 within. Cases III and
IV
(2 scale separation) follow with similar combinations of
Theorem 1 and Lemmas 1 and 2.
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Numerical Results
We consider the problem with
We are computing
2 1 1,
1 2 1
A F
,
,
100EXACT FEM E I
EXACT E I
U UError
U
(An exact solution is available.)
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22
We will be comparing the following methods:
The h version on a uniform mesh, with p = 1, 2, 3
The p version on a single element, with p = 1, 2, …
The hp version on the 5 element (variable) mesh,
with p = 1, 2, …
The h version on a Shishkin mesh, with p = 1, 2, 3
The h version on an exponentially graded mesh, with
p = 1, 2, 3, where the mesh points are given by
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23
1(2 1) ln 1 , 0, ,
2
21 exp
(2 1)
j
sjx p j M
M
sp
0
M
j j jjx x x
with
0 1
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24
100
101
102
103
10-3
10-2
10-1
100
101
102
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 0.4, = 1
slope -3
slope -2
slope -1
h-ver, unif mesh, p = 1
h-ver, unif mesh, p = 2
h-ver, unif mesh, p = 3
p-ver, 1 elem
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25
100
101
102
103
104
10-3
10-2
10-1
100
101
102
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 0.01, = 0.1
slope -0.9
slope -1.9
slope -2.8
h-ver, unif mesh, p = 1
h-ver, unif mesh, p = 2
h-ver, unif mesh, p = 3
p-ver, 1 elem
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26
100
101
102
103
104
10-1
100
101
102
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 10-7/2 3 10-4, = 0.01
slope -1.5slope -0.7
h-ver, unif mesh, p = 1
h-ver, unif mesh, p = 2
h-ver, unif mesh, p = 3
p-ver, 1 elem
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27
100
101
102
103
10-2
10-1
100
101
102
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 10-7/2 3 10-4, = 0.01
slope -0.7
slope -0.87slope -1
h-ver, unif mesh, p = 1
p-ver, 1 elem
hp-ver, 5 elem
h-ver, exp mesh, p = 1
h-ver, Shishkin mesh, p = 1
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28
100
101
102
103
10-3
10-2
10-1
100
101
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 10-7/2 3 10-4, = 0.01
slope -1
slope -2
slope -3
slope -0.87
slope -1.65
slope -2.22
h-ver, unif mesh, p = 1
h-ver, unif mesh, p = 2
h-ver, unif mesh, p = 3
h-ver, Shishkin mesh, p = 1
h-ver, Shishkin mesh, p = 2
h-ver, Shishkin mesh, p = 3
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100
101
102
103
10-3
10-2
10-1
100
101
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 10-7/2 3 10-4, = 0.01
h-ver, unif mesh, p = 1
h-ver, unif mesh, p = 2
h-ver, unif mesh, p = 3
h-ver, Shishkin mesh, p = 1
h-ver, Shishkin mesh, p = 2
h-ver, Shishkin mesh, p = 3
hp-ver, 5 elem
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100
101
102
103
10-3
10-2
10-1
100
101
102
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 10-5, = 10-3
slope -1
slope -0.9
slope -0.65
h-ver, unif mesh, p = 1
p-ver, 1 elem
hp-ver, 5 elem
h-ver, exp mesh, p = 1
h-ver, Shishkin mesh, p = 1
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101
102
10-3
10-2
10-1
100
101
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
hp-version, 5 elements
3 10-4, = 0.01
= 10-5, = 10-3
= 10-6, = 10-4
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Example 2:
2 2
1
2 1 1,
2cos / 4 2.2
02, (0) (1)
010 1
x
x
x xA
x e
ef u u
x
(An exact solution is NOT available, hence we use a
reference solution.)
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33
101
102
103
10-2
10-1
100
101
102
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
= 10-7/2 3 10-4, = 0.01
slope -1
slope -0.87
hp ver, 5 elem
h ver, Shishkin mesh, p=1
h ver, exp mesh, p=1
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101
102
103
10-2
10-1
100
101
102
= 10-5, = 10-3
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
slope -0.87
slope -1
hp ver, 5 elem
h ver, Shishkin mesh, p=1
h ver, exp mesh, p=1
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35
102
10-3
10-2
10-1
100
101
102
DOF
Perc
enta
ge R
ela
tive E
rror
in t
he E
nerg
y N
orm
hp-version, 5 elements
3 10-4, = 0.01
= 10-5, = 10-3
= 10-6, = 10-4
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Closing Remarks
• The hp FEM on the Spectral Boundary Layer Μesh
yields robust exponential convergence in the energy
norm, for the entire range 0 ε μ 1.
• The regularity theory developed was crucial in the
proof of the main approximation result.
• Extending the current approach to systems with more
equations, while conceptually straight forward, appears
to be cumbersome.
• Systems of two equations of convection-diffusion
type are currently being investigated.