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An Example in Two Dimensions 287
equations, which are briefly introduced in section 13.4. However, the theoretical study
of convergence can be quite involved even for simple problems.
Setting up an approximation scheme is not an easy task when Ω is not a square. In this
situation, one can try to map Ω into a square and, after change of variables, discretize
the new differential problem. Another approach, sometimes combined with the previous
one, is to use a domain-decomposition method. The reader interested in this kind
of applications can find a good list of references in the books of canuto, hussaini,
quarteroni and zang(1988), boyd(1989).
13.3 Hints for the implementation
Besides the standard techniques used to numerically solve (13.2.3), other algorithms,
related to the specific structure of the matrix D ⋆ n, n ≥ 2, can be devised and imple-
mented. First, let us introduce some notations. The tensor product of two n × n
matrices W ≡ {wij} 1≤i≤n
1≤j≤n
matrix
(13.3.1) W ⊗ Z :=
and Z ≡ {zij} 1≤i≤n
1≤j≤n
⎡
w11Z w12Z · · ·
⎤
w1nZ
⎢ w21Z
⎢ .
⎢ .
⎣
w22Z
.
· · ·
⎥
w2nZ ⎥
⎥.
.
⎥
⎦
wn1Z wn2Z · · · wnnZ
If W and Z are invertible, then we have the relation
(13.3.2) (W ⊗ Z) −1 = W −1 ⊗ Z −1 .
is defined to be the n 2 × n 2 block
Let În denote the (n − 1) × (n − 1) identity matrix. Thus, we obtain the following
expression:
(13.3.3) D ⋆ n = În ⊗ Ân + Ân ⊗ În,