Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 98

It turns out that it is much better to use an element-oriented approach. For
every element T ∈ T h , we find the additive contribution from (8.1) to the stiffness
matrix. If every element contains exactly s nodes, this requires finding an s × s
submatrix. We transform the triangle T under consideration to the reference triangle
T ref . Let F : T ref → T, ξ → Bξ + x 0 be the corresponding linear mapping. Then
the contribution of T is given by the integral

µ(T )
µ(T ref )

a kl (B − 1 ) k k (B − 1 ) l l ∂ k N i ∂ l N j dξ. ( 8 . 2 )

T ref

k,l
k ,l

Here µ(T ) is the area of T . After transformation, every function in the nodal basis
coincides with one of the normed shape functions N 1 ,N 2 ,...,N s on the reference
triangle. These are listed in Table 4 for linear and quadratic elements. 6 For the
model problem 4.3, using a right triangle T (with right angle at point number 1),
we get

1 1

2 (u 1 − u 3 ) 2 ,
where u i is the coefficient of u in the N i expansion. This gives

2 (u 1 − u 2 ) 2

a(u, u) | T = +

2

− 1 − 1

1
2

a(ψ i ,ψ j ) | T = − 11

− 1 1

for the stiffness matrix on the element level. For linear elements, it is also easy
to find the so-called mass matrix whose elements are (ψ i ,ψ j ) 0 ,T . For an arbitrary
triangle,

211
121
112

.

µ(T )
12

(ψ i ,ψ j ) 0 ,T = ( 8 . 3 )

For differential equations with variable coefficients, the evaluation of the inte-
grals (8.2) is usually accomplished using a Gaussian quadrature formula for multiple

6 To avoid indices, in Table 4 we have written ξ and η instead of ξ 1 and ξ 2 .
In addition, we note that for the quadratic triangular elements, the basis functions
N 1 ,N 2 and N 3 can be replaced by the corresponding nodal basis functions of linear elements.

The coefficients in the expansion i = 1 z i N i then have a different meaning: z 1 ,z 2 and z 3
are still the values at the vertices, but z 4 ,z 5 and z 6 become the deviations at the midpoints
of the sides from the linear function which interpolates at the vertices.
This basis is not a purely nodal basis, although the correspondence is very simple.
However, it has two advantages: we get simpler integrands in (8.2), and the condition
number of the system matrix is generally lower (cf. hierarchical bases).