Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 77

The Bramble-Hilbert Lemma

First we obtain an error estimate for interpolation by polynomials. We begin by
establishing the result for all domains which satisfy the hypotheses of the imbedding
theorem. Later we shall apply it primarily to reference elements, i.e., on convex
triangles and quadrilaterals.

2 be a domain with Lipschitz continuous boundary which
satisfies a cone condition. In addition, let t ≥

6.2 Lemma. Let ⊂ R

2 , and suppose z 1 ,z 2 ,...,z s are

1 )/ 2 prescribed points in ¯

= t(t +

s : such that the interpolation operator I :

H t → P t − 1 is well defined for polynomials of degree ≤ t − 1 . Then there exists a

constant c = c(, z 1 ,...,z s ) such that

for all u ∈ H t (). u − Iu t ≤ c | u | t ( 6 . 2 )

Proof. We endow H t () with the norm

s

||| v ||| : =| v | t +

1 | v(z i ) | , i =

and show that the norms ||| · ||| and · t are equivalent. Then (6.2) will follow from

u − Iu t ≤ c ||| u − Iu |||
= c | u − Iu | t +

s

1 | (u − I u)(z i ) |
= c | u − Iu | t = c | u | t .

i =

Here we have made use of the fact that Iu coincides with u at the interpolation
points, since D α Iu = | α |= t .

One direction of the proof of equivalence of the norms is simple. By Re-
marks 3.4, the imbedding H t → H 2 → C 0

0 for all

is continuous. This implies

| v(z i ) |≤ c v t for i = 1 , 2 ,...,s,

and thus + cs) v t .

Suppose now that the converse

||| v ||| ≤ ( 1

for all v ∈ H t ()

v t ≤ c ||| v ||| ( 6 . 3 )

fails for every positive number c . Then there exists a sequence (v k ) in H t () with

1
k ,k =

v k t = 1 , ||| v k ||| ≤ 1 , 2 ,... .