In numerical analysis, the Bramble-Hilbert** **lemma, named after James H. Bramble and Stephen R. Hilbert, bounds the error of an approximation of a function by a polynomial of order at most in terms of derivatives of of order . Both the error of the approximation and the derivatives of are measured by norms on a bounded domain in .

Theorem. Over a sufficiently domain , there exists a constant such thatfor every where and denote the norm and semi-norm of the Sobolev space .

This is similar to classical numerical analysis, where, for example, the error of linear interpolation can be bounded using the second derivative of . However, the Bramble-Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of are measured by more general norms involving averages, not just the maximum norm.

Additional assumptions on the domain are needed for the Bramble-Hilbert lemma to hold. Essentially, the boundary of the domain must be “reasonable”. For example, domains that have a spike or a slit with zero angle at the tip are excluded. Lipschitz domains are reasonable enough, which includes convex domains and domains with boundary.

The main use of the Bramble-Hilbert lemma is to prove bounds on the error of interpolation of function by an operator that preserves polynomials of order up to , in terms of the derivatives of of order . This is an essential step in error estimates for the finite element method. The Bramble-Hilbert lemma is applied there on the domain consisting of one element.

*Proof*. Let and let , , be a basis of the dual space of . Using the Hahn-Banach extension theorem, there exist continuous linear forms over the space , again denoted , , such that for any , we have , , if and only if . We will show that there exists a constant such that

for every . We assume for a moment the above inequality holds, we show that the theorem follows.

Indeed, given any function , let such that

.

we then have

which becomes

where the last inequality comes from the fact that any th derivatives of equals zero.

We now prove by contradiction. Then there exists a sequence of functions such that for any we have and

.

Since the sequence is bounded in , there exists a subsequence, again denoted by and a function such that

.

This fact comes from the Rellich-Kondrachov Theorem for and from the Arzelà-Ascoli Theorem for . Since

and

and the fact that the space is complete we conclude that the sequence also converges in the space to . The limit of this sequence is such that

for very multi-index such that and thus for all multi-index with . With the connectedness of , it follows from distribution theory that the function is a polynomial of degree less than or equal to . Thus, it follows from

that

so that we conclude that , from the properties of the linear forms . But this contradicts the equality

for all .

From the proof above one can easily obtain the following variant

Theorem. Over a sufficiently domain , there exists a constant and a polynomial such thatfor every where and denote the norm and semi-norm of the Sobolev space .

We also have the following corollary

Corollary. Over a sufficiently domain , there exists a constant such thatfor any

.

See also: http://en.wikipedia.org/wiki/Bramble-Hilbert_lemma

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