1

A GENERAL AND SYSTEMATIC THEORY OF

DISCONTINUOUS GALERKIN METHODS

Ismael HerreraIsmael HerreraUNAM MEXICOUNAM MEXICO

2

THEORY OF

PARTIAL DIFFERENTIAL EQUATIONS

IN DISCONTINUOUS FNCTIONS

A SYSTEMATIC FORMULATION OF

DISCONTINUOUS GALERKIN METHODS

MUST BE BASED ON THE

3

I.- ALGEBRAIC THEORY OF

BOUNDARY VALUE PROBLEMS

4

NOTATIONS

1

2

"Trial and test functions " :

D is linear space of 'trial' functions

D is linear space of 'test' functions

"Bilinear functionals" :

P, B, etc., are bilinear functionals defined on

1 2

1 2

1 2: :*

D D

Pu,w and Bu,w , u,w D D

"Functional - valued operators" :

Having the bilinear functionals P and B, is equivalent

to having a linear 'functional - valued operators',

P D D and B

1 2*D D

5

BASIC DEFINITIONS

1 2 1 2

*

1 2

: :

, 0, 0

:

, 0, 0

* *

B

*2

"Boundary Operator" :

B D D is a boundary operator for P D D when

Pu w w N Pu

"TH - completeness" :

D is TH - complete for P D D when

Pu w w Pu

E

E

6

2

1

1 1

.

.

*

"Boundary Value Problem" :

Let B be a boundary operator for P, and f,g D

The BVP consists in finding u D such that

Pu f and Bu g

"Compatible Data" :

When there are u D & u D such

that

.

P B

f Pu and g Bu

"Existence of solution" :

When the data is 'compatible' and

u - u N N

7

NORMAL DIRICHLET BOUNDARY OPERATOR

1 1 1

1

1

* 0

0, *

2 2

R

RP B

RP B

Auxiliary Subspaces

N* w D P B w D

D D P w w N D

Remark :

u u N N u u D

That is

N N D

B is a

'normal Dirichlet bo

v v,

1 RP B

for P, when

D N N

undary operator'

8

2 2

*

*1 2

,

' '

0,

* *

"Consistent Data" :

The data f,g D D , of the BVP are said to

be consistent when they are 'compatible' and

f - g,w w N

Theorem.- When B : D D is a 'normal Dirichlet

boundary

*

1 2 , operator' for P : D D then the BVP

possesses a solution if and only if the data is

.

'consistent'

EXISTENCE THEOREM

9

II.- BOUNDARY VALUE PROBLEMS

FORMULATED IN

DISCONTINUOUS FUNCTION SPACES

10

PIECEWISE DEFINED FUNCTIONS

1 1 1 1 2 2 1 2

1

... ...

,...,

E E

E

D D D and D D D

u u u

1,..., E

Σ

11

PIECEWISE DEFINED OPERATORS

1 2 1 2

1 2 1 2

1

: :

: :

, , ,

* *i i i i i i

* *

E

i i ii

"Locally Defined Operators " :

P D D and B D D

i = 1,...,E

"Piecewise Defined Operators " :

P D D and B D D

Pu w Pu w and Bu w

1

,E

i i ii

B u w

12

SMOOTH FUNCTIONS

1 1 2 2

* *1 2 1 2

* *1 2 1 2

: :

: : ,

1).- D D and D D are linear subspaces whose members

are said to be 'smooth'

2).- B D D and P D D are restrictions

of B D D and P D D

13

2 1 1

1

,*

"Boundary Value Problem with Prescribed Jumps (BVPJ)" :

Given f,g D and u D find u D such that

Pu f, Bu g and u - u D

with B boundary operator for P.

"Compatible Data" :

It is assumed

1 1

1

*2 2

.

, 0,

that there are functions u D & u D

such that f Pu , g Bu and u - u D

"Consistent Data"

f - Pu w w N D D

14

*1 2

*1 2

2

:

:

2).

1 1

ASSUMPTIONS :

1).- B D D is a 'normal Dirichlet boundary

operator' for P D D

D is TH - complete for P and for B

3).- 'Range invariance'.

P D P D

Then the BVP

J possesses a solution if and only if

the data is . 'consistent'

EXISTENCE THEOREM

for the BVPJ

15

III.- ELLIPTIC EQUATIONS

OF ORDER 2m

16

SOBOLEV SPACE OF

PIECEWISE DEFINED FUNCTIONS

12

1

2

,, ,1

:

ˆ ...

:

ˆ

ˆ,

p p pE

E

pp

p

Definition

H H H

Metric

Then H is a Hilbert space.

v v

17

RELATION BETWEEN SOBOLEV SPACES

1

1

ˆ

ˆ

ˆ

0, , 0,..., 1,

p p

p p

p

p

The spaces H and H are related by

H H

Furthermore : For p 1, if H , then

on Σ for p implies Hn

v

vv

18

THE BVPJ OF ORDER 2m

21 2

2 2

.

ˆ ˆ,

, ,

,

m

m m

j j

The set of 'smooth functions' :

D D D H

Given , of order 2m, and u H find u H such that

u f in each = 1,...,E

B u g j

L

L

2 .

, on , 0,1,..., 2 1

m

rr

r

= 1,...,m -1, on

with the additional condition that u - u D H

This latter restriction is equivalent to the ' ' :

u j r m

n

Here

jump conditions

, on , 0,1,..., 2 1r

rr

u j r m

n

19

EXISTENCE OF SOLUTION FOR THE ELLIPTIC BVPJ

2m

A).- 'Normal Dirichlet BVP'´:

Under standard assumptions for the elliptic differential

operators and boundary operators [Lions and Magenes, 1972],

the BVP formulated in D H , enjoys this property.

0

p

B).- The 'range invariance' assumption :

P D P D

is fulfilled because the range of is H .

REMARK.- Property B) is not satisfied when D H

and p > 2m.

L

20

IV.- GREEN´S FORMULAS IN

DISCONTINUOUS FIELDS

“GREEN-HERRERA FORMULAS (1985)”

21

FORMAL ADJOINTS

1 2 2 1: :* *

*

Two operators, P D D and Q D D are

said to be formal adjoints when S P - Q* is a

boundary operator for P, while S is a boundary

operator for Q

22

1 2 2 1: :* *

*

* *

Let P D D and Q D D be formal

adjoints. Then, the equation

P - B = Q* -C

is said to be a Green's formula, when

B,-C decomposes S P - Q* = B - C .

GREEN’S FORMULA FOR THE BVP

23

1 2 2 1

*

: :* *

*

* *

J

Let P D D and Q D D be formal

adjoints. Then, the equation

P - B - J = Q* -C K

is said to be a Green - Herrera formula, when

B,J,-C ,-K decomposes S P - Q*, and

N

1 2K= D and N = D

GREEN’S FORMULA FOR THE BVPJ

24

* ,

, , * , ,

, , * , ,

, , * , ,

,

w u u w u w

Boundary operators

u w n u w u w on

Jump and average operators

- u w n u w u w on

u w u w n and u w u w n

Pu w w udx

L L D

D =B C

D =J K

J D K D

L

, * , *

, , , * , * ,

, , , * , * ,

Q u w u wdx

Bu w u w dx C u w u w dx

Ju w u w dx K u w u w dx

L

B C

J K

A GENERAL GREEN-HERRERA FORMULA FOR

OPERATORS WITH CONTINUOUS COEFFICIENTS

25

WEAK FORMULATIONS OF THE BVPJ

"

"

"

* * *

Starting with a Green - Herrera formula

P - B - J = Q C K

Weak formulation in terms of the data" :

P - B - J u f g j

Weak formulation in terms of the complement

*

, ,

*

* * *

ary information" :

Q* -C - K u f g j

"Classification of the information"

Data of the BVPJ : Pu, Bu, Ju

Complementary information : Q u C u K u

26

V.- APPLICATION TO DEVELOP

FINITE ELEMENT METHODS

WITH OPTIMAL FUNCTIONS

(FEM-OF)

27

GENERAL STRATEGY

• A target of information is defined. This is denoted by “S*u”

• Procedures for gathering such information are constructed from which the numerical methods stem.

28

EXAMPLE

SECOND ORDER ELLIPTIC

• A possible choice is to take the ‘sought information’ as the ‘average’ of the function across the ‘internal boundary’.

• There are many other choices.

29

CONJUGATE DECOMPOSITIONS

J J

*

A 'Conjugate Decomposition' fulfills :

1).-

K = S + R and J = S + R

2).- S u is the 'sought information'

3).- When the 'sought information' is given, the equation

J R P - B - R u = f - g - j

defines well - posed 'local' problems.

*

1

0

S R S J R J

P

J P R P

REMARKS.- Here and in what follows :

A).- j j j with j S u and j R u

B).- The function u is 'defined' by

P B R u f g j & S u

C).- u D is such that Ju = j, and

D)

.- Homogeneous boundary conditions will be assumed

30

OPTIMAL FUNCTIONS

1

2

0

0

JB J P B R

T Q C R

Optimal Base Functions

O D P B R N N N

Optimal Test Functions

O w D Q C R w N N N

v v

31

THE STEKLOV-POINCARÉ APPROACH

ˆ ˆ

ˆ, ,

* *B

J J P S B

Let u O , then S u S u if and only if

- S u w = S u w - j ,w , w O

THE TREFFTZ-HERRERA APPROACH

1

*

ˆ ˆ

ˆ,

* *

T

Let u D , then S u S u if and only if

- S u w = f - j,w , w O

THE PETROV-GALERKIN APPROACH

ˆ ˆ

ˆ ˆ, , , ,

* *B

*J J P S T

Let u O , then S u S u if and only if

- S u w - S u w f j w = S u w - j ,w , w O

32

ESSENTIAL FEATURE OFFEM-OF METHODS

B T

B T

The linear spaces of 'optimal functions', O and O ,

are replaced by finite dimensional spaces, O and O ,

whose members are approximate 'optimal' base and

test functions.

33

THREE VERSIONS OF FEM-OF

• Steklov-Poincaré FEM-OF

• Trefftz-Herrera FEM-OF

• Petrov-Galerkin FEM-OF

34

FEM-OF HAS BEEN APPLIED TO DERIVE NEW AND MORE EFFICIENT

ORTHOGONAL COLLOCATION METHODS:

TH-COLLOCATION

•TH-collocation is obtained by locally applying orthogonal collocation to construct the ‘approximate optimal functions’.

35

CONCLUSION

The theory of discontinuous Galerkin methods, here

presented, supplies a systematic and general

framework for them that includes a Green formula

for differential operators in discontinuous functions

and two ‘weak formulations’. For any given

problem, they permit exploring systematically the

different variational formulations that can be

applied. Also, designing the numerical scheme

according to the objectives that have been set.

36

MAIN APPLICATIONS OF THIS THEORY OF dG METHODS, thus far.

• Trefftz Methods. Contribution to their foundations and improvement.

• Introduction of FEM-OF methods.

• Development of new, more efficient and general collocation methods.

• Unifying formulations of DDM and preconditioners.