Definitions

# Bramble-Hilbert lemma

In mathematics, particularly numerical analysis, the Bramble-Hilbert lemma, named after James H. Bramble and Stephen R. Hilbert, bounds the error of an approximation of a function $textstyle u$ by a polynomial of order at most $textstyle m-1$ in terms of derivatives of $textstyle u$ of order $textstyle m$. Both the error of the approximation and the derivatives of $textstyle u$ are measured by $textstyle L^\left\{p\right\}$ norms on a bounded domain in $textstyle mathbb\left\{R\right\}^\left\{n\right\}$. This is similar to classical numerical analysis, where, for example, the error of linear interpolation $textstyle u$ can be bounded using the second derivative of $textstyle u$. However, the Bramble-Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of $textstyle u$ 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 continuously differentiable boundary.

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

## The one dimensional case

Before stating the lemma in full generality, it is useful to look at some simple special cases. In one dimension and for a function $textstyle u$ that has $textstyle m$ derivatives on interval $textstyle left\left( a,bright\right)$, the lemma reduces to

$inf_\left\{vin P_\left\{m-1\right\}\right\}biglVert u^\left\{left\left( kright\right) \right\}-v^\left\{left\left( kright\right) \right\}bigrVert_\left\{L^\left\{p\right\}left\left( a,bright\right) \right\}leq Cleft\left( mright\right) left\left(b-aright\right) ^\left\{m-k\right\}biglVert u^\left\{left\left( mright\right) \right\}bigrVert_\left\{L^\left\{p\right\}left\left(a,bright\right) \right\},$

where $textstyle P_\left\{m-1\right\}$ is the space of all polynomials of order at most $textstyle m-1$.

In the case when $textstyle p=infty$, $textstyle m=2$, $textstyle k=0$, and $textstyle u$ is twice differentiable, this means that there exists a polynomial $textstyle v$ of degree one such that for all $textstyle xinleft\left( a,bright\right)$,

$leftvert uleft\left( xright\right) -vleft\left( xright\right) rightvert leq Cleft\left(b-aright\right) ^\left\{2\right\}sup_\left\{left\left( a,bright\right) \right\}leftvert u^\left\{primeprime \right\}rightvert.$

This inequality also follows from the well-known error estimate for linear interpolation by choosing $textstyle v$ as the linear interpolant of $textstyle u$.

## Statement of the lemma

Suppose $textstyle Omega$ is a bounded domain in $textstyle mathbb\left\{R\right\}^\left\{n\right\}$, $textstyle ngeq1$, with boundary $textstyle partialOmega$ and diameter $textstyle d$. $textstyle W_\left\{p\right\}^\left\{k\right\}\left(Omega\right)$ is the Sobolev space of all function $textstyle u$ on $textstyle Omega$ with weak derivatives $textstyle D^\left\{alpha\right\}u$ of order $textstyle leftvert alpharightvert$ up to $textstyle k$ in $textstyle L^\left\{p\right\}\left(Omega\right)$. Here, $textstyle alpha=left\left( alpha_\left\{1\right\},alpha_\left\{2\right\},ldots,alpha_\left\{n\right\}right\right)$ is a multiindex, $textstyle leftvert alpharightvert =$ $textstyle alpha_\left\{1\right\}+alpha_\left\{2\right\}+cdots+alpha_\left\{n\right\}$ and $textstyle D^\left\{alpha \right\}$ denotes the derivative $textstyle alpha_\left\{1\right\}$ times with respect to $textstyle x_\left\{1\right\}$, $textstyle alpha_\left\{2\right\}$ times with respect to $textstyle alpha_\left\{2\right\}$, and so on. The Sobolev seminorm on $textstyle W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)$ consists of the $textstyle L^\left\{p\right\}$ norms of the highest order derivatives,

and

$leftvert urightvert _\left\{W_\left\{infty\right\}^\left\{m\right\}\left(Omega\right)\right\}=max_\left\{leftvert alpharightvert =m\right\}leftVert D^\left\{alpha\right\}urightVert _\left\{L^\left\{infty\right\}\left(Omega\right)\right\}$

$textstyle P_\left\{k\right\}$ is the space of all polynomials of order up to $textstyle k$ on $textstyle mathbb\left\{R\right\}^\left\{n\right\}$. Note that $textstyle D^\left\{alpha\right\}v=0$ for all $textstyle vin P_\left\{m-1\right\}$. and $textstyle leftvert alpharightvert =m$, so $textstyle leftvert u+vrightvert _\left\{W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)\right\}$ has the same value for any $textstyle vin P_\left\{k-1\right\}$.

Lemma (Bramble and Hilbert) Under additional assumptions on the domain $textstyle Omega$, specified below, there exists a constant $textstyle C=Cleft\left(m,Omegaright\right)$ independent of $textstyle p$ and $textstyle u$ such that for any $textstyle uin W_\left\{p\right\}^\left\{k\right\}\left(Omega\right)$ there exists a polynomial $textstyle vin P_\left\{m-1\right\}$ such that for all $textstyle k=0,ldots,m$,

$leftvert u-vrightvert _\left\{W_\left\{p\right\}^\left\{k\right\}\left(Omega\right)\right\}leq Cd^\left\{m-k\right\}leftvert urightvert _\left\{W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)\right\}.$

## The original result

The lemma was proved by Bramble and Hilbert under the assumption that $textstyle Omega$ satisfies the strong cone property; that is, there exists a finite open covering $textstyle left\left\{ O_\left\{i\right\}right\right\}$ of $textstyle partialOmega$ and corresponding cones $textstyle \left\{C_\left\{i\right\}\right\}$ with vertices at the origin such that $textstyle x+C_\left\{i\right\}$ is contained in $textstyle Omega$ for any $textstyle x$ $textstyle inOmegacap O_\left\{i\right\}$.

The statement of the lemma here is a simple rewriting of the right-hand inequality stated in Theorem 1 in . The actual statement in is that the norm of the factorspace $textstyle W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)/P_\left\{m-1\right\}$ is equivalent to the $textstyle W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)$ seminorm. The $textstyle W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)$ norm is not the usual one but the terms are scaled with $textstyle d$ so that the right-hand inequality in the equivalence of the seminorms comes out exactly as in the statement here.

In the original result, the choice of the polynomial is not specified, and the value of constant and its dependence on the domain $textstyle Omega$ cannot be determined from the proof.

## A constructive form

An alternative result was given by Dupont and Scott under the assumption that the domain $textstyle Omega$ is star-shaped; that is, there exists a ball $textstyle B$ such that for any $textstyle xinOmega$, the closed convex hull of $textstyle left\left\{ xright\right\} cup B$ is a subset of $textstyle Omega$. Suppose that $textstyle rho _\left\{max\right\}$ is the supremum of the diameters of such balls. The ratio $textstyle gamma=d/rho_\left\{max\right\}$ is called the chunkiness of $textstyle Omega$.

Then the lemma holds with the constant $textstyle C=Cleft\left( m,n,gammaright\right)$, that is, the constant depends on the domain $textstyle Omega$ only through its chunkiness $textstyle gamma$ and the dimension of the space $textstyle n$. In addition, $v$ can be chosen as $v=Q^\left\{m\right\}u$, where $textstyle Q^\left\{m\right\}u$ is the averaged Taylor polynomial, defined as

$Q^\left\{m\right\}u=intlimits_\left\{B\right\}T_\left\{y\right\}^\left\{m\right\}uleft\left( xright\right) psileft\left( yright\right) dx,$

where

$T_\left\{y\right\}^\left\{m\right\}uleft\left( xright\right) =sumlimits_\left\{k=0\right\}^\left\{m-1\right\}sumlimits_\left\{leftvert alpharightvert =k\right\}frac\left\{1\right\}\left\{alpha!\right\}D^\left\{alpha\right\}uleft\left( yright\right) left\left(x-yright\right) ^\left\{alpha\right\}$

is the Taylor polynomial of degree at most $textstyle m-1$ of $textstyle u$ centered at $textstyle y$ evaluated at $textstyle x$, and $textstyle psigeq0$ is a function that has derivatives of all orders, equals to zero outside of $textstyle B$, and such that

$intlimits_\left\{B\right\}psi dx=1.$

Such function $textstyle psi$ always exists.

For more details and a tutorial treatment, see the monograph by Brenner and Scott . The result can be extended to the case when the domain $textstyle Omega$ is the union of a finite number of star-shaped domains, which is slightly more general than the strong cone property, and other polynomial spaces than the space of all polynomials up to a given degree.

## Bound on linear functionals

This result follows immediately from the above lemma, and it is also called sometimes the Bramble-Hilbert lemma, for example by Ciarlet . It is essentially Theorem 2 from .

Lemma Suppose that $textstyle ell$ is a continuous linear functional on $textstyle W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)$ and $textstyle leftVert ellrightVert _\left\{W_\left\{p\right\}^\left\{m\right\}\left(Omega \right)^\left\{^\left\{prime\right\}\right\}\right\}$ its dual norm. Suppose that $textstyle ellleft\left( vright\right) =0$ for all $textstyle vin P_\left\{m-1\right\}$. Then there exists a constant $textstyle C=Cleft\left( Omegaright\right)$ such that

$leftvert ellleft\left( uright\right) rightvert leq CleftVert ellrightVert _\left\{W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)^\left\{^\left\{prime\right\}\right\}\right\}leftvert urightvert _\left\{W_\left\{p\right\}^\left\{m\right\}\left(Omega\right)\right\}.$