The Bramble-Hilbert lemma

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 that

for 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.