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