1

A GENERAL EFFECTIVE PROCEDURE FOR

COMBINING COLLOCATION AND DOMAIN

DECOMPOSITION METHODS

Ismael Herrera* and Robert Yates**Ismael Herrera* and Robert Yates***UNAM and **Multisistemas de Computo*UNAM and **Multisistemas de Computo

MEXICOMEXICO

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THE PROBLEM

• The main technical difficulty stems from the fact that the standard collocation method (orthogonal spline collocation: OSC) yields non-symmetric matrices, even for formally symmetric differential operators.

Combining collocation and DDM presents difficulties that must be overcome

3

SOLUTION OF THE PROBLEM

• In recent years new collocation methods have been introduced which yield symmetric matrices when the differential operators are formally symmetric . Generically they are known as TH-collocation.

• TH-collocation combines orthogonal collocation with a special kind of Finite Element Method: FEM-OF.

New collocation methods

4

STRUCTURE OF THIS TALK

This talk is divided into two parts:

1. Finite Element Method with Optimal

Functions (FEM-OF).

2. TH-collocation

5

NOTATIONS

6

PIECEWISE DEFINED FUNCTIONS

1 ,..., Eu u u

1,..., E

Σ

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THE BOUNDARY VALUE PROBLEM WITH PRESCRIBED JUMPS (BVPJ)

2 1,*

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

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

Pu f, Bu g and Ju = j

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GREEN´S FORMULAS IN

DISCONTINUOUS FUNCTIONS(GREEN-HERRERA FORMULAS,1985)

9

*

" ´

* *

Green s Formulas for the BVPJ"

Green - Herrera formulas have the general form :

P - B - J = Q - C K

" ´

* *

Green s Formulas for the BVP"

Green's formulas have the general form :

P - B = Q - C

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* ,

, , * , ,

, , * , ,

, , * , ,

,

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 ud

L L D

D =B C

D =J K

J D K D

L

, * , *

, , , * , * ,

, , , * , * ,

x 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

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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

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FINITE ELEMENT METHOD

with OPTIMAL FUNCTIONS

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

FEM-OF are procedures for gathering such information.

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CONJUGATE DECOMPOSITIONS

J J

*

1).- A pair of decompositions are introduced :

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.

* 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).- Homogeneous boundary conditions w

ill be assumed

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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

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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

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ESSENTIAL FEATURES 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.

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THREE VERSIONS OF FEM-OF

1

, ,

, ,

B

P B

* * *P T

POINCARÉ - STEKLOV FEM - OF : Seek for u O such that

P - B - J u w = f - Pu ,w - P - B - J u w , w O

TREFFTZ - HERRERA FEM - OF : Seek for u D such that

Q - C - K u w = f - Pu ,w - P - B - J u w , w O

PETROV GALERKIN FEM - O

, ,

,

B

* * *

P T

F : Seek for u O such that

P - B - J u w Q - C - K u w =

f - Pu ,w - P - B - J u w , w O

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EXAMPLE

SECOND ORDER ELLIPTIC

*

* ,

,

,

* ,

n n n

n n

u a u bu cu and w a w b w cw

w u u w u w

u w a u w w u buw

u w a u w u a w b w

u w a w b w u w

L L

L L D

D

J

K na u

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* ,

* ,

n n

n

The 'average' on the internal boundary

u w a w b w u, on

u w w a u, on , on

S

R

A POSSIBLE CHOICE OF THE ‘SOUGHT INFORMATION’

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CONJUGATE DECOMPOSITIONS

:

, ,

, , ,

, , ,

n nJ J n

J J

Define

u w a u w and u w u a w b w

Then

u w u w u w

w u w u w u

S R

J S R

K S R

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THE SYMMETRIC POSITIVE CASE

, ,* * *

B T

The 'optimal' base and test functions are the same.

The bilinear form

P - B - J w Q - C - K w

is symmetrical and positive definite on O O .

v v

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TH-COLLOCATION

• This is obtained by locally applying orthogonal collocation to construct the approximate optimal functions.

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SECOND ORDER ELLIPTIC EQUATIONS

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*

0, 1,...,

0,

0,

0, 1,...,

0,

0,

i

B

i

T

a b c

w a w b w cw

Optimal base functions

in , i E

O on

on

Optimal test functions

w in , i E

w O w on

w on

v v v v

v

v v

v

L

L *

L

L

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CONSTRUCTION OF THE OPTIMAL FUNCTIONS

• An optimal function is uniquely defined when its ‘trace’ is given on Σ.

• Piecewise polynomials, up to a certain degree, are chosen for the traces on the internal boundary Σ.

• Then the well-posed local problems are solved by orthogonal collocation.

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Support of an ‘Optimal Function’

( , )i jx y

CONSTRUCTION BY ORTHOGONAL COLLOCATION

Cubic-Cubic: Four Collocation Points

1 1

, 0,1

0 0 0

1 1 0

2 0 1

* *

,

( ) ( )

( ) ( )

( ) ( )

, ;

ij

ij

ij

ij

ij ij i j

i j

i j

i j

ij

w b x y C H x H y

b H x H y

b H x H y

b H x H y

x y Gaussian points in = 1,..,4

* *( , )x yCollocation at each

ij

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COMPARISON WITH ‘OSC’

• Steklov-Poincaré FEM-OF yields the same solution as

OSC. However, now the system-matrix is positive definite

for differential systems that are symmetric and positive.

• Trefftz-Herrera FEM-OF yields the same order of

accuracy as OSC, although its solution is not necessarily

the same. The system-matrix is positive definite for

differential systems that are symmetric and positive.

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Support of an ‘Optimal Function’

( , )i jx y

CONSTRUCTION BY ORTHOGONAL COLLOCATIONLinear-Quadratic (One collocation point)

* *( , )x yCollocation at each

ij

1 1

* *

,

1,...,4

,

ij i j ij i i j j

ij

w l x l y C l x l x l y l y

x y Gaussian point in

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THE BILINEAR FORM

,

* * * , , ,

0

B T

P B J w

a w b w c w dx

Q C K w w O O

It is positive definite when b and c 0.

v

v v v

v v

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TH-COLLOCATION FOR

ELASTOSTATIC PROBLEMS OF ANISOTROPIC

MATERIALS AND ITS PARALLELIZATION

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*:

0, 1,...,

0,

0,

0, 1,...,

0,

0,

i

B

i

B

C

Optimal base functions

in , i E

O on

on

Optimal test functions

w in , i E

w O w on

w on

L L

L

L

v v v

v

v v

v

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CONSTRUCTION OF THE OPTIMAL FUNCTIONS

• The displacement fields are chosen to be piecewise polynomials, up to a certain degree, on the internal boundary, Σ.

• Then the well-posed local problems are solved by orthogonal collocation.

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THE BILINEAR FORM

,

: :

* * * , , ,

B T

P B J w

w C dx

Q C K w w O O

It is positive definite.

v

v

v v

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ISOTROPIC MATERIALS

,

2 :

* * * , , ,

B T

P B J w

w w dx

Q C K w w O O

It is positive definite.

v

v v

v v

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*

0, 1,...,

0,

0,

0, 1,...,

0,

0,

i

B

i

B

Optimal base functions

in , i E

O on

on

Optimal test functions

w in , i E

w O w on

w on

L L

L

L

v v v v

v

v v

v

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CONCLUSIONS

For any linear differential equation or system of such equations, TH-collocation supplies a new and more effective manner of using orthogonal collocation in combination with DDM. It has attractive features such as:

1. Better structured matrices,

2. The approximating polynomials on the internal boundary and in the element interiors can be chosen independently,

3. The number of collocation points can be reduced.