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79a) p=k+2 p+k even N (0,2) akl = l+ [q(q + 1) − l(l + 1)akq ] . 79b) q=l+2 q+l even These last two expressions represent the expansions of ∂ 2 uN/∂x2 and ∂ 2 uN/∂y 2, respectively, in terms of the trial functions. 0 10 −3 Maximum Error 10 −6 10 −9 10 Legendre tau 2nd−order −12 10 8 9 10 11 12 N 13 14 15 16 Fig. 5. Maximum errors for the Poisson problem for Legendre tau and secondorder ﬁnite-diﬀerence schemes 24 1. 77). An eﬃcient scheme for the solution of these equations is provided in Sect. 1.

Hereafter, a function u deﬁned in (0, 2π) will be called periodic if u(0+ ) and u(2π − ) exist and are equal. , max |u(x) − PN u(x)| → 0 as N → ∞ . 1 The Fourier System 43 (b) If u is of bounded variation on [0, 2π], then PN u(x) converges pointwise to (u(x+ ) + u(x− ))/2 for any x ∈ [0, 2π] (here u(0− ) = u(2π − )). (c) If u is continuous and periodic, then its Fourier series does not necessarily converge at every point x ∈ [0, 2π]. A full characterization of the functions for which the Fourier series is everywhere pointwise convergent is not known.

2 Some Examples of Spectral Methods 21 For several values of N , we denote by uN the G-NI solution (N is the polynomial degree) and by up (p = 1, 2, 3) the (piecewise-polynomial) ﬁniteelement solution corresponding to a subdivision in subintervals of equal size. In all cases, N + 1 denotes the total number of nodal values. In Fig. 4 (left) we plot the maximum error of the solution, while on the right we plot the p p absolute error of the boundary ﬂux |(ν du dx (1) + βu (1)) − g| for p = 1, 2, 3, N .