# Bramble-Hilbert's Lemma

In mathematics , especially in numerical analysis , the **Bramble-Hilbert lemma** , named after James H. Bramble and Stephen R. Hilbert , estimates the error in approximating a function by a polynomial of the maximum order using the -th order derivatives of from. Both the approximation error as well as the derivatives of are norms on a limited area in measured. In classical numerical analysis, this corresponds to an error bound using the second derivative of with linear interpolation of . However, the Bramble-Hilbert lemma also applies in higher dimensions , and the approximation error and the derivatives of can be measured by more general norms, namely not only in the maximum norm , but also in averaged norms.

Additional regularity assumptions on the boundary of the domain are required for the Bramble-Hilbert lemma. Lipschitz continuity of the edge is sufficient for this, in particular the lemma applies to convex areas and areas.

The main application of Bramble-Hilbert's lemma is the proof of error bounds with the help of the derivatives up to the -th order for the error when approximating by an operator that receives at most polynomials of the order . This is an essential step in proving error estimates for the finite element method . The Bramble-Hilbert lemma is applied there to the field that consists of one element.

## formulation

It is a restricted area with a Lipschitz rim and diameter . Further be arbitrary and .

In the Sobolev space , the semi-norm is used

The Bramble-Hilbert lemma says that for each there is a polynomial whose degree is at most , so that the inequality

is satisfied with a constant .

## Web links

- Raytcho D. Lazarov:
*Bramble-Hilbert Lemma .*In:*Encyclopaedia of Mathematics*(English).