---
tag: K4P0R83G1Q
kind: statement
role: lemma
name: Bramble–Hilbert lemma
msc: [65N15, 65N30, "46E35"]
uses: [M4CEZ5K9HP, 4KX81Q85DT, 07GNX8GCYP, C3EC148VZD]
about: [M4CEZ5K9HP]
sources:
  - url: https://en.wikipedia.org/wiki/Bramble%E2%80%93Hilbert_lemma
same_as:
  wikidata: Q1816909
  wikipedia: Bramble–Hilbert lemma
---

## Statement

Let $\Omega \subset \R^n$ be a bounded open set of diameter $d$.
Assume $\Omega$ is star-shaped with respect to every point of an open ball $B \subset \Omega$ of radius $\rho > 0$.
Let $m \ge 1$ be an integer and $1 \le p \le \infty$.
Let $u$ be in the [[Sobolev space|M4CEZ5K9HP]] $W^{m,p}(\Omega)$ and let $0 \le k \le m$.
Let $\abs{u}_{W^{k,p}(\Omega)}$ be the [[seminorm|4KX81Q85DT]] formed from the $L^p$ norms of the [[weak derivatives|C3EC148VZD]] of $u$ of order exactly $k$.
Then there is a constant $C$, depending only on $n$, $m$, $p$ and $d / \rho$, with the following property.
For every $u \in W^{m,p}(\Omega)$ there is a [[polynomial|07GNX8GCYP]] $q$ of degree less than $m$ such that
$$\abs{u - q}_{W^{k,p}(\Omega)} \le C \, d^{\,m - k} \, \abs{u}_{W^{m,p}(\Omega)} \quad \text{for } k = 0, 1, \ldots, m.$$

## Remarks

One can take for $q$ the averaged Taylor polynomial of $u$ of order $m$.
It is the Taylor polynomial of degree $m - 1$ about a point $y$, averaged over $y \in B$ with a smooth weight.
The remainder of this averaged Taylor formula is an integral operator applied to the derivatives of order $m$ of $u$, and its bounds give the estimate.
The scaling $d^{\,m-k}$ follows by mapping $\Omega$ to a set of diameter $1$.

The right side involves only the seminorm of order $m$, with no lower-order terms.
Every polynomial of degree less than $m$ has $\abs{q}_{W^{m,p}} = 0$, so such a bound can only control $u$ up to these polynomials, which is why $q$ appears.

Example: for $m = 1$ the lemma gives a constant $c$ with $\norm{u - c}_{L^p(\Omega)} \le C d \, \norm{\nabla u}_{L^p(\Omega)}$, a Poincaré-type inequality.

In finite element analysis, the lemma is applied on a reference element to the error $u - I u$ of an interpolation operator $I$ that reproduces polynomials of degree less than $m$.
Since $u - I u = (u - q) - I(u - q)$, the error is bounded by the seminorm $\abs{u}_{W^{m,p}}$, and scaling back to an element of size $h$ gives the factor $h^{m-k}$.

## Explanations

### L1 human

A function whose derivatives of order $m$ are small is close to a polynomial of degree less than $m$.
The lemma makes this precise in Sobolev norms, with a constant that only depends on the shape of the domain.
It is the basic tool behind the error estimates of the finite element method.

### L2 human

For a triangle of size $h$ and piecewise linear interpolation, the lemma with $m = 2$ gives errors of order $h^2$ in $L^2$ and of order $h$ in $H^1$, for $u \in H^2$.
The constant depends on $d / \rho$, which is why finite element meshes must avoid very thin elements.
Combined with Céa's lemma, it turns approximation by polynomials into convergence rates of the finite element method.