In mathematics, particularly numerical analysis, the Bramble–Hilbert lemma, named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function
by a polynomial of order at most
in terms of derivatives of
of order
. Both the error of the approximation and the derivatives of
are measured by
norms on a bounded domain in
. This is similar to classical numerical analysis, where, for example, the error of linear interpolation
can be bounded using the second derivative of
. However, the Bramble–Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of
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
by an operator that preserves polynomials of order up to
, in terms of the derivatives of
of order
. 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
that has
derivatives on interval
, the lemma reduces to
![{\displaystyle \inf _{v\in P_{m-1}}{\bigl \Vert }u^{\left(k\right)}-v^{\left(k\right)}{\bigr \Vert }_{L^{p}\left(a,b\right)}\leq C\left(m\right)\left(b-a\right)^{m-k}{\bigl \Vert }u^{\left(m\right)}{\bigr \Vert }_{L^{p}\left(a,b\right)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06005341954204ddeeee6ef9bd4e09a4b4794369)
where
is the space of all polynomials of order at most
.
In the case when
,
,
, and
is twice differentiable, this means that there exists a polynomial
of degree one such that for all
,
![{\displaystyle \left\vert u\left(x\right)-v\left(x\right)\right\vert \leq C\left(b-a\right)^{2}\sup _{\left(a,b\right)}\left\vert u^{\prime \prime }\right\vert .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/209677901f3ff312ba59505e423e3c0823917388)
This inequality also follows from the well-known error estimate for linear interpolation by choosing
as the linear interpolant of
.
Statement of the lemma
[dubious – discuss]
Suppose
is a bounded domain in
,
, with boundary
and diameter
.
is the Sobolev space of all function
on
with weak derivatives
of order
up to
in
. Here,
is a multiindex,
and
denotes the derivative
times with respect to
,
times with respect to
, and so on. The Sobolev seminorm on
consists of the
norms of the highest order derivatives,
![{\displaystyle \left\vert u\right\vert _{W_{p}^{m}(\Omega )}=\left(\sum _{\left\vert \alpha \right\vert =m}\left\Vert D^{\alpha }u\right\Vert _{L^{p}(\Omega )}^{p}\right)^{1/p}{\text{ if }}1\leq p<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/705ead05ea7f50f44366eb5764c50b62b0648140)
and
![{\displaystyle \left\vert u\right\vert _{W_{\infty }^{m}(\Omega )}=\max _{\left\vert \alpha \right\vert =m}\left\Vert D^{\alpha }u\right\Vert _{L^{\infty }(\Omega )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d6d68502693298f09d406182fdc877d09568f2)
is the space of all polynomials of order up to
on
. Note that
for all
. and
, so
has the same value for any
.
Lemma (Bramble and Hilbert) Under additional assumptions on the domain
, specified below, there exists a constant
independent of
and
such that for any
there exists a polynomial
such that for all
![{\displaystyle \left\vert u-v\right\vert _{W_{p}^{k}(\Omega )}\leq Cd^{m-k}\left\vert u\right\vert _{W_{p}^{m}(\Omega )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e929ccb6f2d4a761ac1110dc1538e501830df4)
The original result
The lemma was proved by Bramble and Hilbert [1] under the assumption that
satisfies the strong cone property; that is, there exists a finite open covering
of
and corresponding cones
with vertices at the origin such that
is contained in
for any
.
The statement of the lemma here is a simple rewriting of the right-hand inequality stated in Theorem 1 in [1]. The actual statement in [1] is that the norm of the factorspace
is equivalent to the
seminorm. The
norm is not the usual one but the terms are scaled with
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
cannot be determined from the proof.
An alternative result was given by Dupont and Scott [2] under the assumption that the domain
is star-shaped; that is, there exists a ball
such that for any
, the closed convex hull of
is a subset of
. Suppose that
is the supremum of the diameters of such balls. The ratio
is called the chunkiness of
.
Then the lemma holds with the constant
, that is, the constant depends on the domain
only through its chunkiness
and the dimension of the space
. In addition,
can be chosen as
, where
is the averaged Taylor polynomial, defined as
![{\displaystyle Q^{m}u=\int _{B}T_{y}^{m}u\left(x\right)\psi \left(y\right)\,dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8580e88a3e4bab9c457e8381190000038dc2fe)
where
![{\displaystyle T_{y}^{m}u\left(x\right)=\sum \limits _{k=0}^{m-1}\sum \limits _{\left\vert \alpha \right\vert =k}{\frac {1}{\alpha !}}D^{\alpha }u\left(y\right)\left(x-y\right)^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df2bf34117cb2d4e13af75f65cf2f76a7013153c)
is the Taylor polynomial of degree at most
of
centered at
evaluated at
, and
is a function that has derivatives of all orders, equals to zero outside of
, and such that
![{\displaystyle \int _{B}\psi \,dx=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bda24b3d28aa64ddb88119b2aa78327ab769b882)
Such function
always exists.
For more details and a tutorial treatment, see the monograph by Brenner and Scott [3]. The result can be extended to the case when the domain
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.[2]
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 [4]. It is essentially Theorem 2 from [1].
Lemma Suppose that
is a continuous linear functional on
and
its dual norm. Suppose that
for all
. Then there exists a constant
such that
![{\displaystyle \left\vert \ell \left(u\right)\right\vert \leq C\left\Vert \ell \right\Vert _{W_{p}^{m}(\Omega )^{^{\prime }}}\left\vert u\right\vert _{W_{p}^{m}(\Omega )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7143c7dfa0f38126c74e77a53c47e066d423e3c)
References
- ^ a b c d J. H. Bramble and S. R. Hilbert. Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal., 7:112–124, 1970.
- ^ a b Todd Dupont and Ridgway Scott. Polynomial approximation of functions in Sobolev spaces. Math. Comp., 34(150):441–463, 1980.
- ^ Susanne C. Brenner and L. Ridgway Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 2002. ISBN 0-387-95451-1
- ^ Philippe G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam]. ISBN 0-89871-514-8
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