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form of (22.38) is
1/w
1+r
(22.40a)
ρ=±
2
β
(22.40b)
θ= (1 + s)
2
The original Cartesian coordinates are given by x = ρ(r) cos(θ[s]) and y = ρ(r) sin(θ[s] ).
The polar form is very useful, however, because it shows that ρw becomes a linear function
of the computational coordinate r, and thus is nonsingular in the (r-s) plane.
Similarly, one can choose a triangular element in the x-y (physical) plane. To de¬ne
the element, specify sides of length A and B which intersect at the origin with an angle
β between them, oriented so that the side of length A is parallel to the x-axis. Then a
“desingularizing” mapping is
1/w
(1 ’ s)2 B 2 + (1 + s)2 A2 + 2 (1 ’ s2 )AB cos(β)
1+r
(22.41a)
ρ=
2 4
B sin(β)(1 ’ s)
(22.41b)
θ = arctan
B cos(β)(1 ’ s) + A(1 + s)
Neither (22.40) nor (22.41) is unique; many other choices of mappings are possible. The
point is that if one knows the degree of the corner branch and if one is willing to accept
the extra programming complexity of additional element shapes and mappings, spectral
elements can eliminate corner singularities.


22.11 Prospectus
The geometries of nature and engineering are both complicated, full of arcs and spirals and
web-like networks of beams and curving cylinders. It is far easier to ¬t a complicated shape
22.11. PROSPECTUS 493

by an assemblage of many triangles or curved quadrilaterals than by a single deformed
square. It is for this reason that ¬nite elements have become so popular.
However, as computers grow in speed and power, second order methods become less
and less satisfactory. After all, Sir Robert Southwell and his colleagues de G. Allen and
Vaisey were applying fourth order methods with desktop calculators half a century ago.
The future of science and engineering is likely to be domain decomposition strategies
that use polynomials of medium order ” sixth to eighth degree, say ” on each element.
Nevertheless, the fundamentals of spectral series are independent of N , the degree of
the polynomial, and also independent of whether the domain of the polynomial is the
whole domain of the problem, or merely a subdomain. Like Newtonian mechanics, spec-
tral methods will surely endure rapid, perhaps even revolutionary change. Like Newto-
nian mechanics, what has been described in these twenty-two chapters will not be replaced,
but merely extended.
Chapter 23

Books and Reviews
“Aun Aprendo” ” “I am still learning”

”“ Francisco de Goya (1746-1828), written on a sketch drawn in his old age



Table 23.1: Books and Reviews on Spectral Methods

Books
Bernardi&Maday(1992) 250 pp. In French; mathematical theory
Boyd(1989,2000) 665 pp. Text & Encyclopedia; elementary mathematical level
Canuto, Hussaini, 500 pp. Half algorithms and practice, half theory
Quarteroni&Zang(1987) Available in paperback and hardcover
Delves&Freeman(1981) 275 pp. Theory of Chebyshev-Galerkin methods and
block-and-diagonal (“Delves-Freeman”) matrix iterations
Finlayson(1972) 410 pp. Low order pseudospectral & Galerkin
lots of examples from ¬‚uids and chemical engineering
Fornberg(1996) 200 pp. Classical pseudospectral; very practical
Available in paperback and hardcover
Fox&Parker(1968) 200 pp. Very readable, but its Galerkin-by-recurrence and
differential-eq.-through-double-integration are now rare
Funaro, D. (1992) 300 pp. Variational and approximation theory plus algorithms;
one chapter on domain decomposition
Funaro, D. (1997) 210 pp. Spectral elements; readable description of the
variational theory; many numerical examples
Gottlieb & Orszag(1977) 150 pp. Mix of examples and stability theory
Guo(1998) 200 pp. Proofs; highly mathematical
Karniadakis&Sherwin(1999) 448 pp. Quadrilateral and triangular spectral elements
Mercier(1989) 200 pp. Translation, without updating, of earlier book in French

Review Articles
Bert&Malik(1996) 28 pp. Pseudospectral as high order ¬nite differences under
the name “differential quadrature”; mechanics applications
Delves(1976) 13 pp. Integral and differential equations; Chebyshev basis
block-and-diagonal (Delves-Freeman) matrix iterations
Fornberg&Sloan(1994) 65 pp. Chebyshev & Fourier single-domain
Givi&Madnia(1993) 44 pp. Spectral methods in combustion
Gottlieb,Hussaini&Orszag(1984) 54 pp. Single-domain pseudospectral
Hussaini,Kopriva&Patera(1989) 31 pp. Chebyshev & Fourier single-domain
Jarraud&Baede(1985) 41 pp. Spectral methods in numerical weather prediction
Maday&Patera(1987) 73 pp. Spectral elements; ¬‚uids
Robson & Prytz(1993) 31 pp. Pseudospectral as the “discrete ordinates” method
physics applications
Fischer&Patera(1994) 44 pp. Spectral elements on massively parallel machines



494
Appendix A

A Bestiary of Basis Functions

“Talk with M. Hermite: he never invokes a concrete image; yet you soon perceive that the
most abstract entities are for him like living creatures.”
” Henri Poincar´ e




A.1 Trigonometric Basis Functions: Fourier Series
Applications: All problems with periodic boundary conditions

x ∈ [’π, π] x ∈ [0, 2 π]
Interval: or

(Because of the periodicity, these two intervals are equivalent.)

General Fourier Series:

∞ ∞
(A.1)
f (x) = a0 + an cos(nx) + bn sin(nx)
n=1 n=1


π
a0 = (1/2π) f (x)dx
’π
π
an = (1/π) f (x) cos(nx)dx
’π
π
(A.2)
bn = (1/π) f (x) sin(nx)dx
’π




495
APPENDIX A. A BESTIARY OF BASIS FUNCTIONS
496

If f (x) is symmetric about x = 0, one needs only the cosine terms, giving a Fourier
cosine series. If f (x) is antisymmetric about the origin, that is, f (x) = ’f (’x) for all x,
then one needs only the sine terms. One should halve the computational interval to avoid
applying redundant collocation conditions, i. e.
ππ Fourier Cosine Series
x ∈ [0, π] x∈ ’ ,
Interval: or
or Fourier Sine Series
22
If f (x) has de¬nite symmetry with respect to both x = 0 and x = π/2, then one needs
only the even or odd terms of a Fourier cosine or sine series as explained in Chapter 8. For
these “quarter-Fourier series” in which one uses only the cosine terms with even n, or the
sine terms of even n, then
“Quarter-Wave” Series with De¬nite Parity
x ∈ [0, π/2]
Interval:
with Respect to Both x = 0 and x = π/2

Collocation Points:

±
(General)
 π i/N i = 1, . . . , 2N

(Symmetric or Antisymmetric) (A.3)
π i/N i = 1, . . . , N
xi =


(Quarter-Wave [Double Symmetry])
πi/(2N ) i = 1, . . . , N
A.2. CHEBYSHEV POLYNOMIALS: TN (X) 497

A.2 Chebyshev Polynomials: Tn (x)


Applications: Any problem whatsoever.
1
ρ(x) ≡ √ (A.4)
Weight:
1 ’ x2
±
Inner Product: 
0 m=n

1
T m Tn
√ (A.5)
π m=n=0
dx =

1 ’ x2 
’1
π/2 m=n=0



(A.6)
Tn (cos[t]) = cos(nt)
Standardization:



|Tn (x)| ¤ 1 for all n, all on [’1, 1] (A.7)
Inequality:



Three-Term Recurrence:

T0 ≡ 1 T1 (x) ≡ x (A.8)
;
Tn+1 (x) = 2 x Tn (x) ’ Tn’1 (x) n≥1 (A.9)



Differentiation rules:

(i) Use the de¬nition of Tn (x) in terms of the cosine, (A.6), and Table E“2 [RECOM-
MENDED because of its simplicity]. For example,

dTn n sin(nt)
(A.10)
=
dx sin(t)
d 2 Tn n2 cos(nt) n cos(t) sin(nt)
=’ (A.11)
+
sin2 (t) sin3 (t)
dx2

where t = arccos(x).

(ii) Use the relationship between the Chebyshev and Gegenbauer polynomials, which
is
dm (m)
Tn (x) = n 2m’1 (m ’ 1)! Cn’m (x) (A.12)
dxm

p’1
n2 ’ k 2
d p Tn n+p
(A.13)
= (±1)
dxp 2k + 1
x=±1 k=0
APPENDIX A. A BESTIARY OF BASIS FUNCTIONS
498

(iii) Compute the Chebyshev series for the derivative from that for u(x) itself. Let
N
dq u (q)
(A.14)
= ak Tk (x)
dxq
k

so that the superscript “q” denotes the coef¬cients of the q-th derivative. These may be
computed from the Chebyshev coef¬cients of the (q ’ 1)-st derivative by the recurrence
relation (in descending order)
(q) (q)
(A.15)
aN = aN ’1 =0

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