Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 62

Fig. 13. A triangulation which is shape regular but not uniform

Since h =

max T ∈ T h T , uniformity is a stronger requirement than shape reg-
ularity. Clearly, the triangulations shown in Figs. 13 and 14 are shape regular,
independent of how many steps of the refinement in the neighborhood of the
boundary or of a reentrant corner are carried out. However, if the number of steps
depends on h , the partitions are no longer uniform.

In practice, we almost always use shape-regular meshes, and very frequently
even uniform ones.

Significance of the Differentiability Properties

In the conforming treatment of second order elliptic problems, we choose finite
elements which lie in H 1 . We shall show that it is possible to use functions which
are continuous but not necessarily continuously differentiable. Thus, the functions
are much less smooth than required for a classical solution of the boundary-value
problem.

In the following, we will always assume unless otherwise indicated that the
partitions satisfy the requirements of 5.1. We say that a function u on satisfies
a given property piecewise provided that its restriction to every element has that
property.

5.2 Theorem. Let k ≥ 1 and suppose is bounded. Then a piecewise infinitely

differentiable function v : ¯

→ R belongs to H k () if and only if v ∈ C k − 1 ( ¯ ).

Proof. It suffices to give the proof for k = 1. For k> 1 the assertion then follows

immediately from a consideration of the derivatives of order k − 1. In addition,

2 .

for simplicity we restrict ourselves to domains in R

(1) Let v ∈ C( ¯ ) , and suppose

M
j =

T ={ T j }

1 is a partition of .For i =

1 , 2,

define w i : → R = ∂ i v(x) for x ∈ , where on the edges

we can take either of the two limiting values. Let φ ∈ C 0 () . Green's formula

piecewise by w i (x) :