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118 3. Finite Element Methods for Linear Elliptic Problems

for  
a 11 ··· a 1,d+1
. .
 . . . . 
 . . . 
B =   . (3.49)
 
a d1 ··· a d,d+1
1 ··· 1
The matrix B is nonsingular due to assumption (3.45); that is, λ(x)=

B −1 x , and hence
1
d

λ i (x)= c ij x j + c i,d+1 for all i =1,... ,d +1 ,
j=1
where C =(c ij ) := B −1 .
ij
Consequently, the λ i are affine-linear, and hence λ i ∈P 1 .The level

surfaces x ∈ K
λ i (x)= µ correspond to intersections of hyperplanes
with the simplex K (see Figure 3.3). The level surfaces for distinct µ 1 and
µ 2 are parallel to each other, that is, in particlular, to the level surface for
µ = 0, which corresponds to the triangle face spanned by all the vertices
apart of a i .

λ = 1 λ = µ
1 2 1
a
. 3
a .
31
. a 23
.
a 1
.
a
12 .
a
2
Figure 3.3. Barycentric coordinates and hyperplanes.

By (3.48), the barycentric coordinates can be defined for arbitrary x ∈ R d
(with respect to some fixed d-simplex K). Then

x ∈ K ⇐⇒ 0 ≤ λ i (x) ≤ 1 for all i =1,... ,d +1 .

x
Applying Cramer’s rule to the system Bλ = ,we get for the ith
1
barycentric coordinate
 
a 11 ··· x 1 ··· a 1,d+1
. . .
1  . . . 


λ i (x)= det  . . .  .
det(B)  
a d1 ··· x d ··· a d,d+1
1 ··· 1 ··· 1