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. ${\displaystyle u}$ ${\displaystyle m-1}$ ${\displaystyle m}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle L^{p}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle L^{p}}$

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. ${\displaystyle C^{1}}$

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. ${\displaystyle m}$ ${\displaystyle m-1}$

formulation

It is a restricted area with a Lipschitz rim and diameter . Further be arbitrary and . ${\displaystyle \textstyle \Omega }$ ${\displaystyle \textstyle \mathbb {R} ^{n}}$ ${\displaystyle \textstyle d}$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle k\in \{0,\ldots ,m\}}$

In the Sobolev space , the semi-norm is used ${\displaystyle W_{p}^{k}(\Omega )}$

${\displaystyle |u|_{W_{p}^{k}(\Omega )}:=\left(\sum \limits _{|\alpha |=k}\|D^{\alpha }u\|_{L^{p}(\Omega )}^{p}\right)^{\frac {1}{p}}.}$

The Bramble-Hilbert lemma says that for each there is a polynomial whose degree is at most , so that the inequality ${\displaystyle u\in W_{p}^{m}(\Omega )}$ ${\displaystyle v}$ ${\displaystyle m-1}$

${\displaystyle |u-v|_{W_{p}^{k}(\Omega )}\leq Cd^{m-k}|u|_{W_{p}^{m}(\Omega )}}$

is satisfied with a constant . ${\displaystyle C=C(m,\Omega )}$

Web links

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