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152 3. Finite Element Methods for Linear Elliptic Problems

ˆ
where ω i = ω i,K =ˆω i | det(B)| and b i = b i,K := F(b i ) are dependent on
K. The positivity of the weights is preserved. Here, again F(ˆx)= Bˆx + d
ˆ
denotes the affine-linear transformation from K to K. The errors of the
quadrature formulas
R

ˆ
ˆ
E(ˆv):= ˆ v(ˆx) dˆx − ˆ ω i ˆv(b i ) ,
ˆ K
i=1
(3.114)
R

E K (v):= v(x) dx − ω i v(b i )
K
i=1
are related to each other by
ˆ
E K (v)= | det(B)|E(ˆv) . (3.115)
The accuracy of a quadrature formula will be defined by the requirement
that for l as large as possible,
ˆ
ˆ
E(ˆp)= 0 for ˆp ∈P l (K)
is satisfied, which transfers directly to the integration over K. A quadrature
formula should further provide the desired accuracy by using quadrature
nodes as less as possible, since the evaluation of the coefficient functions is
often expensive. In contrast, for the shape functions and their derivatives
a single evaluation is sufficient. In the following we discuss some exam-
ples of quadrature formulas for the elements that have been introduced in
Section 3.3.
The most obvious approach consists in using nodal quadrature formu-
ˆ ˆ ˆ
las, which have the nodes ˆa 1 ,..., ˆa L of the reference element (K, P, Σ) as
ˆ
quadrature nodes. The requirement of exactness in P is then equivalent to

ˆ ω i = N i (ˆx) dˆx, (3.116)
ˆ K
so that the question of the validity of (3.113) remains.
ˆ
We start with the unit simplex K defined in (3.47). Here, the weights
of the quadrature formulas can be given directly on a general simplex K:If
the shape functions are expressed by their barycentric coordinates λ i ,the
integrals can be computed by
α 1 !α 2 ! ··· α d+1 ! vol (K)
α 1
λ λ α 2 ··· λ α d+1 (x) dx = (3.117)
1 2 d+1 ˆ
K (α 1 + α 2 + ··· + α d+1 + d)! vol (K)
(see Exercise 3.28).
If P = P 1 (K) and thus the quadrature nodes are the vertices, it follows
that

1
ω i = λ i (x) dx = vol (K) for all i =1,... ,d +1 . (3.118)
K d +1