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An Example in Two Dimensions 289
The scheme (13.3.5)-(13.3.6) is implicit. However, we can invert the block diagonal
matrix −An + ωIn in (13.3.5) by computing once and for all the inverse of the block
− Ân + ω În. Similarly, by writing (−KnAnKn + ωIn) −1 = Kn(−An + ωIn) −1 Kn, we
can deal with (13.3.6). The theory shows that the fastest convergence of the method is
obtained when ω = λn,1 λn,n−1, where λn,1 and λn,n−1 are the smallest and the
largest eigenvalues of − Ân respectively.
To conclude this section, we present the results of a numerical test. In (13.1.1) we
take f(x,y) := 2π, ∀(x,y) ∈ Ω, i.e., a constant charge density in Ω. The corresponding
potential U is plotted in figure 13.3.1. Besides, in table 13.3.1, we report for different n
the value of the approximating polynomial pn at the center of Ω. Chebyshev collocation
nodes have been used. As n increases, these computed quantities converge to a limit,
which is expected to be the potential at the point (0,0).
Figure 13.3.1 - Solution of problem (13.1.1) for f ≡ 2π.