# 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.