Bramble–Hilbert lemma

In mathematics, particularly in numerical analysis, estimates the Bramble - Hilbert lemma, named after James H. Bramble and Stephen R. Hilbert, the error in approximation of a function by a polynomial of maximum order using the first-order derivatives of from. Both the approximation error as well as the derivatives of be measured by standards on a bounded domain in. In classical numerical analysis, this corresponds to an error bound using the second derivative with linear interpolation. However, the Bramble - Hilbert lemma holds also in higher dimensions, and the approximation error and the derivatives of this can be measured by more general standards, not only in the maximum norm, but also in average standards.

Additional regularity assumptions on the edge of the area required for the lemma of Bramble - Hilbert. Lipschitz continuity of the edge is sufficient for this purpose, in particular, the lemma is true for convex domains and domains.

The main application of the lemma of Bramble - Hilbert is the proof of error bound using the derivatives up to the nth order of the error in approximation by an operator, the polynomials of order will receive a maximum. This is an essential step in the proof of error estimates for the finite element method. The lemma of Bramble Hilbert there is applied to the area, which consists of an element.

Formulation

It is a bounded domain with Lipschitz boundary and in diameter. Next is arbitrary and.

On the Sobolev space, use the seminorm

The lemma of Bramble - Hilbert states that, for every a polynomial exists, whose degree is at most, so that the inequality

Is fulfilled by a constant.