On the finite element approximation of 4 th order singularly - - PowerPoint PPT Presentation

On the finite element approximation

• f 4th order singularly perturbed

eigenvalue problems

Christos Xenophontos Department of Mathematics and Statistics University of Cyprus joint work with D. Savvidou (UCY) and H.-G. Roos (TU-Dresden)

2

The Model Problem

( )

2 (4)( )

( ) ( ) ( ) ( ) in (0, ( ) 1) u x x u x x u u x x I       − + = =

Find such that (0) (1) (0) (1) u u u u   = = = =

4

( ) ( ) , u x C I     Singularly perturbed 4th order eigenvalue problem:

2

where ε  (0, 1] is a given small parameter and α(x), β(x) ≥ 0, are given sufficiently smooth functions.

The Model Problem

( )

2 (4)( )

( ) ( ) ( ) ( ) in (0, ( ) 1) u x x u x x u u x x I       − + = =

Find such that (0) (1) (0) (1) u u u u   = = = =

4

( ) ( ) , u x C I     Singularly perturbed 4th order eigenvalue problem:

3

Remarks: [MOSER, 1955]

3

• The problem is self-adjoint.

Remarks: [MOSER, 1955]

3

• The problem is self-adjoint.
• For all positive eigenvalues λk(ε), we have

lim ( ) (0)

k k 

  

→

= Remarks: [MOSER, 1955]

3

• The problem is self-adjoint.
• For all positive eigenvalues λk(ε), we have

lim ( ) (0)

k k 

  

→

= Remarks: [MOSER, 1955] where λk(0) are the eigenvalues of the reduced/limiting

• problem. If λk(0) is real, then so is λk(ε)

3

• The problem is self-adjoint.
• For all positive eigenvalues λk(ε), we have

lim ( ) (0)

k k 

  

→

= Remarks: [MOSER, 1955] where λk(0) are the eigenvalues of the reduced/limiting

• problem. If λk(0) is real, then so is λk(ε)
• λk(ε) can be expanded as a power series in ε.

4

Theorem: [MOSER, 1955]

, , S BL BL k k k k

u u u u

+ −

= + +

Each eigenfunction uk can be decomposed as where denotes the smooth part, denotes the left boundary layer and denotes the right boundary layer. Moreover, for n = 0, 1, 2, …

S k

u

, BL k

u

+ , BL k

u

−

4

, , S BL BL k k k k

u u u u

+ −

= + +

Each eigenfunction uk can be decomposed as where denotes the smooth part, denotes the left boundary layer and denotes the right boundary layer. Moreover, for n = 0, 1, 2, …

S k

u

, BL k

u

+ , BL k

u

−

( )

( ) , 1 /

( ) ,

n BL n x k k

u x C e 





+ − −



( )

( ) , 1 (1 )/

( )

n BL n x k k

u x C e 





− − − −



( )

( ) ( )

,

n S k k

u x C 

Theorem: [MOSER, 1955]

5

Variational Formulation

( ) ( )

2

, ,

k k k

B u v u v v H I  =  

( )

2

, , , + , B u v u v u v u v        = + where is the usual L2(Ι) inner product and

,  

and such that

( ) ( )

 

2 2

: (0) (0) (1) (1)

k

u H I u H I u u u u    =  = = = = Find

k

 

6

We seek

( )

, ,

h h k k h

B u v u v v V  =  

Discretization

( )

2

,

• s. t.

h h k h k

u V H I    

6

We seek

( )

2

,

• s. t.

h h k h k

u V H I    

We define the energy norm as

( )

, ,

h h k k h

B u v u v v V  =  

Discretization

2 2 2

2 2 2 2 2 2 ( ) ( ) ( )

( )

E L I L I L I

u u u u u H I    = + +  

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We seek We define the energy norm as

( )

, ,

h h k k h

B u v u v v V  =  

Discretization

and we have

( )

2 2

, ( )

E

B u u u u H I    

2 2 2

2 2 2 2 2 2 ( ) ( ) ( )

( )

E L I L I L I

u u u u u H I    = + +  

( )

2

,

• s. t.

h h k h k

u V H I    

In order to define the finite element space Vh let be an arbitrary mesh on I = (0, 1) and set  

1

... 1

N

x x x  = =    =

7

( )

1 1

, , , 1,...,

j j j j j j

I x x h x x j N

− −

= = − =

In order to define the finite element space Vh let be an arbitrary mesh on I = (0, 1) and set  

1

... 1

N

x x x  = =    =

7

( )

1 1

, , , 1,...,

j j j j j j

I x x h x x j N

− −

= = − =

With the space of polynomials on (α, β) of degree less than or equal to p  2N + 1, we define

( , )

p

P  

In order to define the finite element space Vh let

( )

 

2

: ( ), 1, ,

j

h p j I

V u H I u P I j N =   =

be an arbitrary mesh on I = (0, 1) and set  

1

... 1

N

x x x  = =    =

7

( )

1 1

, , , 1,...,

j j j j j j

I x x h x x j N

− −

= = − =

With the space of polynomials on (α, β) of degree less than or equal to p  2N + 1, we define

( , )

p

P  

Let be an arbitrary partition of the interval (a, b) and suppose that for a sufficiently smooth function f(x), x(a, b), the values are given. Then, there exists a unique polynomial , called the Hermite interpolant of f, given by

 

N i i

x

=

8

Definition:

( ) , ( )

i i i i

f x y f x y   =  = 

2 1( , ) N I

f P a b

+



1, , ( )

( ) ( ( ))

i i i N I i i

y y H f x x H x

=

= + 



where, with Li(x) the Lagrange polynomial of degree N associated with node xi ,

2 2 0, 1,

( ) 1 2( ) ( ) ( ) , ( ) ( ) ( )

i i i i i i i i

dL H x x x x L x H x x x L x dx   = − − = −    

9

Theorem: Let u  C 2n+2([a, b]) and let be a mesh on [a, b], with maximum mesh size h and with N a multiple of n. If uI is the piecewise Hermite interpolant of u, having degree at most 2n+1 on each subinterval [xi-1, xi], i = 1,…, N then

 

N i i

x

=

 =

9

Theorem:

( )

( ) 2 2 (2 2) ( ) ( )

, 0,1,...,2 1

k I n k n L I L I

u u Ch u k n

 

+ − +

−  = +

 

N i i

x

=

 = Let u  C 2n+2([a, b]) and let be a mesh on [a, b], with maximum mesh size h and with N a multiple of n. If uI is the piecewise Hermite interpolant of u, having degree at most 2n+1 on each subinterval [xi-1, xi], i = 1,…, N then

9

Theorem:

( )

( ) 2 2 (2 2) ( ) ( )

, 0,1,...,2 1

k I n k n L I L I

u u Ch u k n

 

+ − +

−  = +

In our setting

( )

( ) 1 ( 1) ( ) ( )

, 0,1,...,

k I p k p L I L I

u u Ch u k p

 

+ − +

−  =

 

N i i

x

=

 = Let u  C 2n+2([a, b]) and let be a mesh on [a, b], with maximum mesh size h and with N a multiple of n. If uI is the piecewise Hermite interpolant of u, having degree at most 2n+1 on each subinterval [xi-1, xi], i = 1,…, N then

10

Definition: Exponentially graded mesh With N > 4 a multiple of 4 we define

, ,

( ) ln(1 4 ) , [0,1/ 4 1/ ] 1 exp ( 1)

p p

t C t t N C p

 

   = − −  −   = − −   +  

10

Definition: Exponentially graded mesh

3 /4 /4 1 /4 1

( 1) ( / ) , 0,1,..., / 4 1 ( / 4 1) , / 4...,3 / 4 2 / 2 1 ( 1) , 3 / 4 1,...,

N N j N

p j N j N x x x x j N j N N N N j p j N N N      

− −

 + = −    −   = + − + =    +     −   − + = +      

With N > 4 a multiple of 4 we define

, ,

( ) ln(1 4 ) , [0,1/ 4 1/ ] 1 exp ( 1)

p p

t C t t N C p

 

   = − −  −   = − −   +  

10

Definition: Exponentially graded mesh

3 /4 /4 1 /4 1

( 1) ( / ) , 0,1,..., / 4 1 ( / 4 1) , / 4...,3 / 4 2 / 2 1 ( 1) , 3 / 4 1,...,

N N j N

p j N j N x x x x j N j N N N N j p j N N N      

− −

 + = −    −   = + − + =    +     −   − + = +      

With N > 4 a multiple of 4 we define

, ,

( ) ln(1 4 ) , [0,1/ 4 1/ ] 1 exp ( 1)

p p

t C t t N C p

 

   = − −  −   = − −   +  

11

Lemma: [X., CMAM 2017] Let uBL denote either boundary layer and let be its Hermite interpolant based on the exponential mesh. Then

( )

( ) 1 ( 1 ) ( )

, 0,1,...,

k I k p k BL BL L I

u u C N k p 



− − + −

−  =

I BL

u and

2

1/2 1 ( ) I p BL BL H I

u u C N  −

− +

− 

11

Lemma: [X., CMAM 2017]

( )

( ) 1 ( 1 ) ( )

, 0,1,...,

k I k p k BL BL L I

u u C N k p 



− − + −

−  =

and

2

1/2 1 ( ) I p BL BL H I

u u C N  −

− +

− 

Using the above lemma and assuming N < ε –1, we establish Let uBL denote either boundary layer and let be its Hermite interpolant based on the exponential mesh. Then

I BL

u

( )

( ) 1 ( 1 ) ( )

, 0,1,...,

k I k p k L I

u u C N k p 



− − + −

−  =

12

( )

( ) 1 ( 1 ) ( )

, 0,1,...,

k I k p k L I

u u C N k p 



− − + −

−  =

2

1/2 1 ( ) I p H I

u u C N  −

− +

− 

12

( )

( ) 1 ( 1 ) ( )

, 0,1,...,

k I k p k L I

u u C N k p 



− − + −

−  =

2

1/2 1 ( ) I p H I

u u C N  −

− +

− 

1 I p E

u u CN − + − 

12

( )

( ) 1 ( 1 ) ( )

, 0,1,...,

k I k p k L I

u u C N k p 



− − + −

−  =

2

1/2 1 ( ) I p H I

u u C N  −

− +

− 

Sketch of proof: We use the decomposition of u into a smooth part and two boundary layers. The layers are handled by the previous lemma and the smooth part by the assumption N < ε –1. ฀

12

1 I p E

u u CN − + − 

13

Proposition: For all h ≤ h0, with h0 independent of ε, there holds

( )

2 2

1

h p k k k k

C h   

−

  +

13

Proposition: For all h ≤ h0, with h0 independent of ε, there holds Sketch of proof: We use the classical techniques found in

[Strang & Fix, 1973], utilizing the Ritz projection Rw of

• nto Vh :

2 0 ( )

w H I 

( )

2 2

1

h p k k k k

C h   

−

  +

13

Proposition: For all h ≤ h0, with h0 independent of ε, there holds Sketch of proof: We use the classical techniques found in

[Strang & Fix, 1973], utilizing the Ritz projection Rw of

• nto Vh :

2 0 ( )

w H I  ( , )

h

B w Rw v v V − =  

( )

2 2

1

h p k k k k

C h   

−

  +

13

Proposition: For all h ≤ h0, with h0 independent of ε, there holds Sketch of proof: We use the classical techniques found in

[Strang & Fix, 1973], utilizing the Ritz projection Rw of

• nto Vh :

2 0 ( )

w H I  ( , )

h

B w Rw v v V − =  

There holds

1. p E

w Rw Ch − − 

( )

2 2

1

h p k k k k

C h   

−

  +

14

Other tools used include the minimax principle and the fact that the Green’s function associated with our problem is uniformly bounded. Continuity of the bilinear form and Galerkin orthogonality are also utilized. ฀

14

Other tools used include the minimax principle and the fact that the Green’s function associated with our problem is uniformly bounded. Continuity of the bilinear form and Galerkin orthogonality are also utilized. ฀ For the approximation of the eigenfunctions, we have the following, under the assumption that all eigenvalues are distinct.

15

Proposition: Assume that the eigenfunctions and their approximations are normalized and that all eigenvalues are distinct. Then

1 h p k k k E

u u C h − − 

15

Proposition: Assume that the eigenfunctions and their approximations are normalized and that all eigenvalues are distinct. Then Sketch of proof: The main observation is the identity

2

2 ( )

( , )

h h h h k k k k k k k k k L I

B u u u u u u    − − = − + −

1 h p k k k E

u u C h − − 

Then, for h sufficiently small, there holds

16

k h k j

j k         −

Then, for h sufficiently small, there holds

k h k j

j k         −

hence

2 2

2 2 ( ) 2 2( 1) ( )

2(1 )

h h h k k k k k k k E L I p k k k L I

u u u u u Ru C h    

−

−  − + −  + − 

16

฀

17

Numerical Results

We consider the problem

( )

2 (4)( )

( ) ( ) in (0,1) ( )

x

u x e u x xu x I u x     − + = = (0) (1) (0) (1) u u u u   = = = =

No exact solution is available, so for the computations we use a reference solution obtained with twice as many degrees of freedom (DOF).

18

Approximation of the 1st eigenvalue

19

Approximation of the 2nd eigenvalue

Approximation of the 1st eigenfunction

20

Approximation of the 2nd eigenfunction

21

22

Closing Remarks

We considered a 4th order singularly perturbed eigenvalue problem and studied the performance of an h FEM on the Exponentially Graded Mesh.

22

Closing Remarks

We considered a 4th order singularly perturbed eigenvalue problem and studied the performance of an h FEM on the Exponentially Graded Mesh. The derivative of the eigenfunctions features boundary

• layers. Once the layers are resolved, classical results

give us the required convergence (including the ‘doubling effect’ for the eigenvalues).

22

Closing Remarks

We considered a 4th order singularly perturbed eigenvalue problem and studied the performance of an h FEM on the Exponentially Graded Mesh. The derivative of the eigenfunctions features boundary

• layers. Once the layers are resolved, classical results

give us the required convergence (including the ‘doubling effect’ for the eigenvalues). Numerical results corroborate our theoretical findings.