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In ¬nite-element partitionings, the original discrete set is subdivided into subsets , C

each consisting of a distinct set of elements. Given a ¬nite element discretization of the

6

domain , a ¬nite dimensional space of functions over is de¬ned, e.g., functions ”

B

that are piecewise linear and continuous on , and that vanish on the boundary of .

Consider now the Dirichlet problem on and recall that its weak formulation on the ¬nite

element discretization can be stated as follows (see Section 2.3):

£ £x }¤ }¤

6 6

Find such that ¥x

4

5

” t D t t ” t

}

where the bilinear form is de¬ned by

4 ¢sh x

ht

£ £ £

¢¥ ¢ ¢ ¢

£x £ £ £

}¤

¢ ¢

¢ ¢ ¢ ¢

4 t D h D| |

h

t

¦ ¨|

¨|

c| c|

¡ ¡

It is interesting to observe that since the set of the elements of the different ™s are disjoint,

C

}

can be decomposed as

4 ish x

ht

£x £ x C¤4

‚ W

4

} }¤

4 t D t t

c sC

where

£xC £ £

4

}

¢ ¢

4 t D h E|

h

¡

d

In fact, this is a generalization of the technique used to assemble the stiffness matrix from

element matrices, which corresponds to the extreme case where each consists of exactly C

one element.

If the unknowns are ordered again by subdomains and the interface nodes are placed

—&™

™

p w}g3y w

{7 | p˜ 7 { f p uq£ … ¢ 7

}¢ ˜

7|

¥£¤§

¡ !

§ #

"

! §¤ "

…¤"’ hC §

¤

last as was done in Section 13.1, immediately the system shows the same structure,

£¤ £¤ ¨¦

§ ¨¦

§ £¤ ¨¦

§

¥

¤ ¤§ c5 § ¤ §

’

c

c|

c

¤ ¤§

§ ¤ §

¥

¤ § | ¤ § 5 ¤ §¨

¨ ¨

¨

unH4 ¦ j

. . .

.. ˜ ¥

˜

. . .

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D

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