Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 91

7.4 Remark. According to Theorem 6.4, we should get a higher-order error bound
for quadratic triangular elements under the assumption of H 3 -regularity. This ob-
servation is misleading, however, since - except in some special cases - smooth
boundaries are required for H 3 -regularity. But a domain with smooth boundary
cannot be decomposed into triangles, and the usual problems arise along the curved
boundaries (cf. Ch. III, §1).
There is more regularity in the interior of the domain, and in most cases, the
finite element approximations with quadratic or cubic triangles are so much better
than with piecewise linear ones that it is worth the extra effort to use them.
The estimate (7.3) holds for any affine family of triangular elements which
contains the P 1 elements as a subset. Moreover, by Theorem 6.7 analogous results
hold if we use bilinear quadrilateral elements instead of linear triangles. Using the
same arguments as in the proof of the previous theorem, we get

7.5 Theorem. Suppose we are given a set of shape-regular partitions of into
parallelograms. Then the finite element approximation u h by bilinear quadrilateral
elements in S h satisfies

u − u h 1 ≤ ch f 0 . ( 7 . 4 )

L 2 -Estimates
If the polynomial approximation error is measured in the L 2 -norm (i.e., in the H 0 -
norm), then by Theorem 6.4 the order of approximation is better by one power of
h . It is not at all obvious that this property carries over to finite element solutions.
The proof uses the H 2 -regularity a second time, and requires a duality argument
which has been called Nitsche's Trick . We now present an abstract formulation of
it; cf. Aubin [1967] and Nitsche [1968].

7.6 Aubin-Nitsche Lemma. Let H be a Hilbert space with the norm and

the scalar product ( · , · ). Let V be a subspace which is also a Hilbert space under
another norm

|·|

· . In addition, let

V → H be continuous.

Then the finite element solution in S h ⊂ V satisfies

1
| g |

| u − u h |≤ C u − u h sup

g ∈ H

inf
v S h ϕ g − v

( 7 . 5 )

∈

where for every g ∈ H , ϕ g ∈ V denotes the corresponding unique (weak) solution
of the equation

a(w, ϕ g ) = (g, w) for all w ∈ V. ( 7 . 6 )