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overlapping, quadrilateral spectral elements We, e = 1, …,K, which are not fitted to G,

such that ¯W =

K

e=1

We is satisfied.

Each spectral element is mapped to the parent domain D = ?1,1°ø?1,1, where

for each point (x ,h) ? D there exists a point (x(x ,h), y(x ,h)) ? We, using the transfinite

mapping, F, of Schneidesch and Deville 62 such that x = F(x ). The mapping F

is defined by:

F(x ,h) =g1(x )f1(h)+g2(h)f2(x )+g3(x )f2(h)+g4(h)f1(x )

?x1f1(h)f1(x )?x2f1(h)f2(x )?x3f2(h)f2(x )?x4f2(h)f1(x )

(1.7)

where the parameterisations gi map the parent element boundaries ˆGi onto the corresponding

physical element boundaries Gi, gi : ˆGi ?Gi and are given by:

g1(x ) = x(x ,?1)g2(h) = x(1,h)g3(x ) = x(x ,1)g4(h) = x(?1,h) (1.8)

Each gi, i = 1, …,4, is defined by:

gi(x ) =

gi(1)?gi(?1)

2

x +

gi(1)+gi(?1)

gi(1)?gi(?1)

(1.9)

so that the corners of our physical element are given by x1 = g1(?1) = g4(?1), etc.

The so-called blending functions fi are given by:

f1(x ) =

1?x

2

f2(x ) =

1+x

2

(1.10)

Let PN(We) denote the space of all polynomials on We of degree less than or equal to

N and define:

PN(W) :=

f ; f|We ? PN(We)

and

VN :=V ?PN(W).

Define the 1D interface

G = {(x, y) ? W; y = ¯ y}, W1 = {(x, y) ? W; y < ¯ y}, W2 = WW1.
Note that the line y = ¯ y is chosen as the intersection between the boundaries of the two
sub-domains because 0 is a member of the Gauss-Lobatto Legendre grid.
We construct the spectral element interpolant of this function on a uniformly spaced
grid using a single element. The uniformly spaced grid is denoted:
Du =
Mx
k=1
xk?1, xk°ø
My
m