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256 Polynomial Approximation of Differential Equations
Due to the different behavior of U in the two subdomains, the best results are
obtained with an uneven distribution of nodes (n1 > n2). A correct balance of the two
parameters guarantees performances as good as the ones of the single-domain approxi-
mation, with the same degrees of freedom and a lower computational cost. In fact, the
matrices involved in the multidomain approach are block-diagonal, with a number of
blocks equal to the number of subdomains and a bandwidth proportional to the largest
degree used. We can take advantage of this structure for a more efficient numerical
implementation. The details are discussed in section 11.3. In (11.2.23), we explicitly
write the system from the above example with n1 = n2 = 2 (hence γ1 = 3):
(11.2.13)
⎡
⎢
⎣
1 0 0 0 0 0
−4 ˜ d (2)
10 −4 ˜ d (2)
11 −4 ˜ d (2)
12 0 0 0
0 0 1 −1 0 0
τ1 τ2 τ3 τ4 τ5 τ6
0 0 0 −4 ˜ d (2)
10 −4 ˜ d (2)
11 −4 ˜ d (2)
12
0 0 0 0 0 1
⎤⎡
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎢
⎣
pn1,1(θ (n1,1)
0
pn1,1(θ (n1,1)
1
pn1,1(θ (n1,1)
2
pn2,2(θ (n2,2)
0
pn2,2(θ (n2,2)
1
pn2,2(θ (n2,2)
2 )
⎤ ⎡
)
⎥ ⎢
) ⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
) ⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=
⎢
) ⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
) ⎥ ⎢
⎥ ⎢
⎦ ⎣
σ1
⎥
) ⎥
0 ⎥
f(s1) ⎥
) ⎥
⎦
f(θ (n1,1)
1
f(θ (n2,2)
1
where τ1 := −2 ˜ d (2)
20 + 2γ1 ˜ d (1)
20 , τ2 := −2 ˜ d (2)
21 + 2γ1 ˜ d (1)
21 , τ3 := −2 ˜ d (2)
22 + 2γ1 ˜ d (1)
22 ,
τ4 := −2 ˜ d (2)
00 −2γ1 ˜ d (1)
00 , τ5 := −2 ˜ d (2)
01 −2γ1 ˜ d (1)
01 , τ6 := −2 ˜ d (2)
02 −2γ1 ˜ d (1)
02 . We can remove
one unknown by eliminating the third row and summing up the third and the fourth
columns. Similarly, condition (11.2.6) is imposed by defining τ1 := 2 ˜ d (1)
20 , τ2 := 2 ˜ d (1)
21 ,
τ3 := 2 ˜ d (1)
22 , τ4 := −2 ˜ d (1)
00 , τ5 := −2 ˜ d (1)
01 , τ6 := −2 ˜ d (1)
02 , and by setting to zero the
fourth entry of the right-hand side vector.
In the Chebyshev case, relation (11.2.6) is more suitable for implementation since
it is difficult to obtain the exact expression of γk in (11.2.8), or the counterpart of
formula (11.2.12). As an alternative, we can use the condition
σ2
⎤