An hp-FEM for a singularly perturbed problem of reaction ...xenophon/slides/BAIL_2018.pdf · An...
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An hp-FEM for a singularly perturbed
reaction-convection-diffusion problem
with two small parameters
Christos Xenophontos
Department of Mathematics and Statistics
University of Cyprus
joint work with
Irene Sykopetritou (Ph.D. student)
1
► The model problem
► Analytic regularity of the solution
► Variational formulation
► Discretization by an hp FEM – error estimates
► Numerical example
► Closing remarks
Outline
2
The Model Problem
1 2( ) ( ) ( ) ( ) ( ) ( ) in (0,1)
(0) (1) 0
u x b x u x c x u x f x I
u u
Find u(x) such that
2
The Model Problem
1 2( ) ( ) ( ) ( ) ( ) ( ) in (0,1)
(0) (1) 0
u x b x u x c x u x f x I
u u
where 0 < ε1, ε2 << 1 are given constants and b(x), c(x),
f(x) are given (analytic) functions, which satisfy
Find u(x) such that
0 ,n
2
The Model Problem
1 2( ) ( ) ( ) ( ) ( ) ( ) in (0,1)
(0) (1) 0
u x b x u x c x u x f x I
u u
Find u(x) such that
( ) ( ) ( )
, , ,!, ! , !n n n n n n
f f b b c cI I If C n b C n c C n
0 ,n
for some constants , , .b c fC C
where 0 < ε1, ε2 << 1 are given constants and b(x), c(x),
f(x) are given (analytic) functions, which satisfy
3
In addition, we assume there exist constants β, γ, ρ such
that
2( ) 0 , ( ) 0 , ( ) ( ) 02
b x c x c x b x
3
In addition, we assume there exist constants β, γ, ρ such
that
2( ) 0 , ( ) 0 , ( ) ( ) 02
b x c x c x b x
Example: u(x) for
ε1 = 0.0005
ε2 = 0.05
b = c = f = 1
4
There exist constants K, C > 0, independent of ε and n such
that the solution u(x) satisfies
Proposition:
( ) 1 1
1 2( )max , ,
nn n
L Iu CK n
4
There exist constants K, C > 0, independent of ε and n such
that the solution u(x) satisfies
( ) 1 1
1 2( )max , ,
nn n
L Iu CK n
Remark:
This classical differentiability result is sufficient in the so-
called asymptotic range of the (approximating polynomial)
degree p, but not for practical values of p.
Proposition:
5
Let λ0(x), λ1(x) be the solutions of the characteristic equation
and set0 0 1 1
[0,1][0,1]max ( ) , min ( )
xxx x
or equivalently2 2
2 2 1
0,1[0,1]
1
( ) ( ) 4 ( )min
2x
b x b x c x
5
Let λ0(x), λ1(x) be the solutions of the characteristic equation
and set0 0 1 1
[0,1][0,1]max ( ) , min ( )
xxx x
or equivalently2 2
2 2 1
0,1[0,1]
1
( ) ( ) 4 ( )min
2x
b x b x c x
The values of μ0 , μ1 determine the strength of the
boundary layer and since |λ0(x)| < |λ1(x)|, the boundary layer
near x = 1 is stronger.
6
Essentially, there are three regimes [Linß, 2010]:
μ0 μ1
Convection-Diffusion ε1 << ε2 = 1 1 1/ε1
Reaction-Convection-Diffusion ε1 << (ε2)2 << 1 1 /ε2 ε2 /ε1
Reaction-Diffusion 1 >> ε1 >> (ε2)2 1/ (ε1)
1/2 1/ (ε1)1/2
6
Essentially, there are three regimes [Linß, 2010]:
We will focus on the case
μ0 μ1
Convection-Diffusion ε1 << ε2 = 1 1 1/ε1
Reaction-Convection-Diffusion ε1 << (ε2)2 << 1 1 /ε2 ε2 /ε1
Reaction-Diffusion 1 >> ε1 >> (ε2)2 1/ (ε1)
1/2 1/ (ε1)1/2
2
1 2 1
and we anticipate layers of width O(1/ε2) at the left endpoint
and width O(ε2 /ε1) at the right endpoint.
6
Essentially, there are three regimes [Linß, 2010]:
μ0 μ1
Convection-Diffusion ε1 << ε2 = 1 1 1/ε1
Reaction-Convection-Diffusion ε1 << (ε2)2 << 1 1 /ε2 ε2 /ε1
Reaction-Diffusion 1 >> ε1 >> (ε2)2 1/ (ε1)
1/2 1/ (ε1)1/2
2
1 2 1
and we anticipate layers of width O(1/ε2) at the left endpoint
and width O(ε2 /ε1) at the right endpoint.
For simplicity we will assume that ( ) , ( ) .b x b c x c
We will focus on the case
7
Define the stretched variables
2 2 1/ , (1 ) /x x x x
and make the formal ansatz
7
Define the stretched variables
and make the formal ansatz
2, ,2 1 2 ,
0 0
~ / ( ) ( ) ( )BLj BLi
i j i ji j
i j
u u x u x u x
with the functions to be determined., ,, , ,BLBL
i j i ji ju u u
2 2 1/ , (1 ) /x x x x
8
Substituting in the DE and equating like powers of ε1, ε2 we
obtain the following:
8
0,0
,0 1,0
0, 1,
, 2, 1 1,
/
, 1
0, 1
1, 2, 1
i i
j j
i j i j i j
u f c
bu u i
c
u u j
u u bu i jc
Smooth part:
Substituting in the DE and equating like powers of ε1, ε2 we
obtain the following:
,0 ,0
, , , 1
0, 0
, 0, 1
BL BL
i i
BL BL BL
i j i j i j
b u cu i
b u cu u i j
Left boundary layer:
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,0 ,0
, 1 , 1 ,
0, 0
, , 0
BL BL
i i
BL BL BL
i j i j i j
u b u i
u b u cu i j
Left boundary layer:
Right boundary layer:
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,0 ,0
, , , 1
0, 0
, 0, 1
BL BL
i i
BL BL BL
i j i j i j
b u cu i
b u cu u i j
10
There exist constants
independent of ε and n such that
Proposition:
( )
, 1 2( )! ! 0,1,2,...n n n
i j L Iu Cn K i K n
2
( )/
, 2
( )
( )n n i jBL xi j n
i j
L I
u C K i j e
2 1
( )
(1 ) /, 2 1
( )
( ) /
n n i jBLn xi j
i j
L I
u C K i j e
, 0, 0,1,2,...i j n
1 2, , , , , , , , 0C K K C K C K
11
Next, we define for some M
2
2 1 2 ,
0 0
( ) / ( )M M
ji
M i j
i j
u x u x
(Smooth Part)
2,2 1 2
0 0
( ) / ( )M M
jBL BLi
M i j
i j
u x u x
(Left BL)
2, ,2 1 2
0 0
( ) / ( )M MBL BLji
i j i j
i j
u x u x
(Right BL)
11
Next, we define for some M
2
2 1 2 ,
0 0
( ) / ( )M M
ji
M i j
i j
u x u x
(Smooth Part)
2,2 1 2
0 0
( ) / ( )M M
jBL BLi
M i j
i j
u x u x
(Left BL)
2, ,2 1 2
0 0
( ) / ( )M MBL BLji
i j i j
i j
u x u x
(Right BL)
(Remainder)BLBL
M MM Mr u u u u
11
Next, we define for some M
2
2 1 2 ,
0 0
( ) / ( )M M
ji
M i j
i j
u x u x
(Smooth Part)
2,2 1 2
0 0
( ) / ( )M M
jBL BLi
M i j
i j
u x u x
(Left BL)
2, ,2 1 2
0 0
( ) / ( )M MBL BLji
i j i j
i j
u x u x
(Right BL)
BLBL
M MM Mr u u u u
(Remainder)
and we have the decomposition
BLBL
M MM Mu u u u r
12
By construction, u satisfies the BVP.
12
By construction, u satisfies the BVP.
Moreover, we assume that
(otherwise the previous definitions do not make sense)
12 2
2
1, 1M M
(§)
13
Assume (§) holds. Then there exist positive constants K1, K2,
K3, γ, δ, independent of ε and n such that for any n = 0, 1, 2, …
Proposition:
( )
1( )!n n
M L Iu Cn K
2
( )( , )/
2 2( )n
BL dist x In nMu x CK e
2 1
( )
( , ) /23
1
( )
nnBL
dist x InMu x CK e
2 1 22 2
/ /1/2
1( ) ( ) ( )max ,M M ML I L I L I
r r r e e
14
Variational Formulation
1
0,B u v F v v H I
1
0Find such thatu H I
1 2, , , ,B u v u v bu v cu v
where, with the usual L2(Ι) inner product,
,F v f v
,
It follows that the bilinear form is coercive, i.e.
2 1
0, ,E
B u u C u u H I C
15
denotes the energy norm.
2 2 2
1 1, 0,E I Iu u u
where
16
Discretization
As usual, we seek 1
0 s. t.N Nu V H I
,N NB u v F v v V
16
Discretization
As usual, we seek 1
0 s. t.N Nu V H I
,N NB u v F v v V
where the space VN is defined as follows: Let IST = [–1, 1] be
the reference element and Πp(IST) the space of polynomials on
IST , of degree p. With an arbitrary partition of I,
we define
0
N
i ix
16
Discretization
As usual, we seek 1
0 s. t.N Nu V H I
,N NB u v F v v V
where the space VN is defined as follows: Let IST = [–1, 1] be
the reference element and Πp(IST) the space of polynomials on
IST , of degree p. With an arbitrary partition of I,
we define
0
N
i ix
1
0( ) ( ) : ( ( )) ( ), 1,...,p
N j p STV S u H I u Q I j N
1 1( ) (2 )( )j j j j jQ x x x x where
17
Definition: Spectral Boundary Layer Mesh
1 20, and 0 1p
2
1
0, : withpS p S H
define
1
1
1 1 1
0 1 0
0,1 if 1/ 2
0, ,1 ,1 if 1/ 2
p
p p p
For
The polynomial degree is taken to be uniformly p over all
elements.
0 κp/μ0 1– κp/μ1 1
18
The following result from [Schwab, 1998] is the main tool in the
analysis.
2 2
2
2
222 ( 1)
2( ) ( )
22
2 ( 1)
( )( )
( ) ( ) , ( ) ( )
1 ( )!( )
( )!
( )!( )
( )!
p p
s s
p L I L I
s s
p L IL I
u a I a u b I b
p su I u b a u
p p s
p su I u b a u
p s
Theorem: Let I = [a, b]. Then for any u C(I) there exists
Ipu Πp(I) such that, for any 0 s p,
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where u is the exact solution and Ipu is the interpolant on the
Spectral Boundary Layer Mesh.
p
p Eu I u Ce
Proposition: There exist constants C, > 0, depending only
on the input data, such that
Using the previous theorem, we are able to prove the following
19
where u is the exact solution and Ipu is the interpolant on the
Spectral Boundary Layer Mesh.
p
p Eu I u Ce
Proposition: There exist constants C, > 0, depending only
on the input data, such that
Using the previous theorem, we are able to prove the following
Sketch of Proof: Separated into two cases.
20
Case 1: κp(μ1)–1 ≥ ½, asymptotic case, Δ = {0, 1}
20
Case 1: κp(μ1)–1 ≥ ½, asymptotic case, Δ = {0, 1}
2 2
22( 1)
2( ) ( )
11 1 1
1 22
11
2
1 ( )!
( )!
1 ( )! max 1, ,
( )!
1 ( )! 1 , (0, )
( )!
s
p L I L I
ss
ss
p su I u u
p p s
p sCK s
p p s
p sC K s s p
p p s
20
Case 1: κp(μ1)–1 ≥ ½, asymptotic case, Δ = {0, 1}
2 2
22( 1)
2( ) ( )
11 1 1
1 22
11
2
1 ( )!
( )!
1 ( )! max 1, ,
( )!
1 ( )! 1 , (0, )
( )!
s
p L I L I
ss
ss
p su I u u
p p s
p sCK s
p p s
p sC K s s p
p p s
Choosing s = λp, λ(0, 1) at our disposal, we get
2
2 11
2( )
1 ( )!1
( )!
pp
p L I
p pu I u C K p
p p p
21
2
2
( )
p
p L Iu I u Ce
Stirling’s formula allows us to obtain
21
2
2
( )
p
p L Iu I u Ce
Stirling’s formula allows us to obtain
Similarly
2
2
( )
p
p
L I
u I u Cpe
and this completes the proof in the asymptotic case.
22
Case 2: κp(μ0)–1 < 1/2, pre-asymptotic case
22
Case 2: κp(μ0)–1 < 1/2, pre-asymptotic case
The mesh consists of 3 elements and we use the
decomposition
BLBL
M MM Mu u u u r
22
Case 2: κp(μ0)–1 < 1/2, pre-asymptotic case
The mesh consists of 3 elements and we use the
decomposition
BLBL
M MM Mu u u u r
Each term is approximated separately; the remainder is not
approximated since it is exponentially small.
22
Case 2: κp(μ0)–1 < 1/2, pre-asymptotic case
The mesh consists of 3 elements and we use the
decomposition
BLBL
M MM Mu u u u r
Each term is approximated separately; the remainder is not
approximated since it is exponentially small.
The approximation of the smooth part is the same as in
Case 1.
23
For left boundary layer, separate approximations are
constructed on I1 = [0, κp(μ0)–1] and I \ I1.
23
For left boundary layer, separate approximations are
constructed on I1 = [0, κp(μ0)–1] and I \ I1.
Here we use the linear
interpolant of BL
Mu
23
For left boundary layer, separate approximations are
constructed on I1 = [0, κp(μ0)–1] and I \ I1.
Here we use the linear
interpolant of BL
Mu
Here we construct the interpolant using the previous
theorem.
23
For left boundary layer, separate approximations are
constructed on I1 = [0, κp(μ0)–1] and I \ I1.
Here we use the linear
interpolant of BL
Mu
Here we construct the interpolant using the previous
theorem.
Similarly for the boundary layer on the right.
24
where uFEM is the finite element solution and Ipu is the
interpolant, both on the Spectral Boundary Layer Mesh.
p
FEM p Eu I u Ce
Proposition: There exist constants C, > 0, depending only
on the input data, such that
24
where uFEM is the finite element solution and Ipu is the
interpolant, both on the Spectral Boundary Layer Mesh.
Proposition: There exist constants C, > 0, depending only
on the input data, such that
p
FEM p Eu I u Ce
Sketch of Proof: By coercivity and Galerkin orthogonality,
there holds, with ξ := Ipu – uFEM ,
24
where uFEM is the finite element solution and Ipu is the
interpolant, both on the Spectral Boundary Layer Mesh.
Proposition: There exist constants C, > 0, depending only
on the input data, such that
2
,( , ) ( , )pE I
C B B u I u
p
FEM p Eu I u Ce
Sketch of Proof: By coercivity and Galerkin orthogonality,
there holds, with ξ := Ipu – uFEM ,
25
2
,
1 2
( , ) ( , )
, , ,
pE I
p p pI
I I
C B B u I u
u I u b u I u c u I u
25
2
,
1 2
( , ) ( , )
, , ,
pE I
p p pI
I I
C B B u I u
u I u b u I u c u I u
Cauchy-Schwarz
2
2, , ,,, max 1,p pE I I E IE I
I
C b u I u c u I u
25
2
,
1 2
( , ) ( , )
, , ,
pE I
p p pI
I I
C B B u I u
u I u b u I u c u I u
Cauchy-Schwarz
2
2, , ,,, max 1,p pE I I E IE I
I
C b u I u c u I u
For this term, we integrate by parts to get
26
2 2, ,p pI
I
b u I u b u I u
26
2 2, ,p pI
I
b u I u b u I u
Then, we consider each interval in the mesh separately and
we make use of the fact that
1
0 1/ 2p
26
2 2, ,p pI
I
b u I u b u I u
Then, we consider each interval in the mesh separately and
we make use of the fact that
1
0 1/ 2p
Ultimately, we are able to establish
2 2
,
, ,
,
p pI
I
p
E I
b u I u b u I u
Ce
27
p
FEM Eu u Ce
Proposition: There exist constants C, > 0, depending only
on the input data, such that
Proof:
p
FEM FEM p pE E Eu u u I u I u u Ce
28
Numerical Results
We consider the problem with b = c = f = 1 and we
are computing
,
,
100EXACT FEM E I
EXACT E I
u uError
u
29
30
31
32
33
Closing Remarks
• The hp FEM on the Spectral Boundary Layer Μesh
yields robust exponential convergence for all values of
0 ε1 ε2 1.
33
Closing Remarks
• The hp FEM on the Spectral Boundary Layer Μesh
yields robust exponential convergence for all values of
0 ε1 ε2 1.
• We are in the process of studying a two-dimensional
analog:
1 2( , ) ( ) ( ) ( , ) ( , ) in
0 on
uu x y b x c x u x y f x y
x
u