W W W W

£ ¨£ £

c VVT

TT

W

where each represents the coupling between interior nodes and and represent the E£

)5

C C C

coupling between the interface nodes and the nodes interior to . Note that each of these C

matrices has been assembled from element matrices and can therefore be obtained from

contributions over all subdomain that contain any node of . ) C

In particular, assume that the assembly is considered only with respect to . Then the C

assembled matrix will have the structure

¥

#5

C C

˜ a

DC t

£ C C ¦

a

where contains only contributions from local elements, i.e., elements that are in .

C C

a a

Clearly, is the sum of the ™s, C

‚

a a

W

hC

D

c sC

The Schur complement associated with the interface variables is such that

a

5c v £

£

D

‚ a

5 c v C E£

W

D C C

c sC

‚ ‚

a

c

W W

5 vC C £

C C

D

c sC

c sC

‚ a c

W a

5 vC C £ C C

D h

Y

c sC

£

Therefore, if denotes the local Schur complement

C

a

5 c vC C £ C

£

DC tC

then the above proves that,

un ¦ j

‚ ˜ ¥

˜

W

£ £

tC

D

c sC

showing again that the Schur complement can be obtained easily from smaller Schur com-

plement matrices.

˜

Another important observation is that the stiffness matrix , de¬ned above by re- p

stricting the assembly to , solves a Neumann-Dirichlet problem on . Indeed, consider

p p

the problem

¥ ¥ ¥

un! ¦ j

a5

p p p| pz ˜ ¥

˜

D h

£ p ¦p p ¦p

¦

™ —t

7 u pU¢ 7 p z7 yw— y w u

{ {˜p p { p

˜

&

“£§

¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

a }

The elements of the submatrix are the terms where are the basis func- ¥¤ B t C ¤ x p 4

) &¤ t C ¤

)

p

tions associated with nodes belonging to the interface . As was stated above, the matrix p

a

is the sum of these submatrices. Consider the problem of solving the Poisson equa-

p B

tion on with boundary conditions de¬ned as follows: On , the part of the boundary

p

B B

which belongs to , use the original boundary conditions; on the interfaces with )p

p

other subdomains, use a Neumann boundary condition. According to Equation (2.36) seen