# p.p1 Legendre grid. We construct the spectral element

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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

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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