Robust exponential convergence of - UCYxenophon/slides/SPPSystems.pdf · 2016. 7. 17. · Robust...

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)

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

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

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).

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

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

,

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

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.

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:

9

Theorem 1:



( ) 1/2 1,

ma( x ,I)n

n n

IC n

U

9

Theorem 1:



( ) 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

9

Theorem 1:



( ) 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

10

2

( , )/( )ˆ ( )n dist x In nv x C e

10

2


/

, ,

( ) 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

10

Additionally, the second component

satisfies the sharper estimate

ˆˆ of BLv U

2


/

, ,

( ) 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


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

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 .

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

12

Discretization


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:

12

Discretization


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

Then with Πp the space of polynomials of degree p

over [–1, 1], we have

1 2 3 1, , , ,p pspan

13



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

13



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

14

11 1

( ) 1 12 2

j j jx Q x x

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

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.

16

[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

17

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).

18

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

19

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.

20

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.

21

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.)

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

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

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



p-ver, 1 elem

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




p-ver, 1 elem

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




p-ver, 1 elem

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


p-ver, 1 elem

hp-ver, 5 elem

h-ver, exp mesh, p = 1

h-ver, Shishkin mesh, p = 1

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







29

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







hp-ver, 5 elem

30

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


p-ver, 1 elem

hp-ver, 5 elem

h-ver, exp mesh, p = 1


31

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

32

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.)

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

34

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

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

36

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.

Robust exponential convergence of - UCYxenophon/slides/SPPSystems.pdf · 2016. 7. 17. · Robust...

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