Ngô Quốc Anh

April 22, 2010

The Bramble-Hilbert lemma

Filed under: Giải tích 9 (MA5265) — Ngô Quốc Anh @ 14:27

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 u by a polynomial of order at most k in terms of derivatives of u of order k+1. Both the error of the approximation and the derivatives of u are measured by L^p norms on a bounded domain in \mathbb R^n.

Theorem. Over a sufficiently domain \Omega, there exists a constant C(\Omega) such that

\displaystyle\mathop {\inf }\limits_{v \in {P_k}(\Omega )} {\left\| {u - v} \right\|_{{W^{k + 1,p}}(\Omega )}} \leqslant C(\Omega ){\left| u \right|_{{W^{k + 1,p}}(\Omega )}}

for every u \in W^{k+1,p}(\Omega) where \| \cdot \| and | \cdot| denote the norm and semi-norm of the Sobolev space W^{k+1,p}(\Omega).

This is similar to classical numerical analysis, where, for example, the error of linear interpolation u can be bounded using the second derivative of u. However, the Bramble-Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of u 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 C^1 boundary.

The main use of the Bramble-Hilbert lemma is to prove bounds on the error of interpolation of function u by an operator that preserves polynomials of order up to k, in terms of the derivatives of u of order k+1. 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.

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