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}$

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

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