Bramble–Hilbert lemma explained

style u by a polynomial of order at most style m -1 in terms of derivatives of style u of order style m . Both the error of the approximation and the derivatives of style u are measured by style L ^{p} norms on a bounded domain in style R ^{n} . This is similar to classical numerical analysis, where, for example, the error of linear interpolation style u can be bounded using the second derivative of style u . However, the Bramble–Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of style 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

style u by an operator that preserves polynomials of order up to style m -1 , in terms of the derivatives of style u of order style 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 (or, in some superconvergence results, a small number of elements).

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

style u that has style m derivatives on interval style \left( a,b\right) , the lemma reduces toinf _{v\in P _{m-1}} l \Vert u ^{\left(} - v ^{\left(} r \Vert _{L ^{p} \left( a,b\right)} \leq C\left( m,k\right) \left( b - a\right) ^{m-k} l \Vert u ^{\left(} r \Vert _{L ^{p} \left( a,b\right)} foreachinteger k\leq m andextendedreal p\geq 1 ,

where

style P _{m-1} is the space of all polynomials of degree at most style m -1 and f ^{(k)} indicatesthe k th derivative of a function f .

In the case when

style p =inf ty , style m =2 , style k =0 , and style u is twice differentiable, this means that there exists a polynomial style v of degree one such that for all style x\in\left( a,b\right) ,\left\vert u\left( x\right) - v\left( x\right) \right\vert \leq C\left( b - a\right) ^{2} \sup _{\left(} \left\vert u ^{\prime\prime} \right\vert.

This inequality also follows from the well-known error estimate for linear interpolation by choosing

style v as the linear interpolant of style u .

Statement of the lemma

Suppose

style \Omega is a bounded domain in style R ^{n} , style n\geq 1 , with boundary style \partial\Omega and diameter style d . style W ^{k(\Omega)} _{p} is the Sobolev space of all function style u on style \Omega with weak derivatives style D ^{\alpha} u of order style \left\vert \alpha\right\vert up to style k in style L ^{p(\Omega)} . Here, style \alpha = \left( \alpha _{1,\alpha} _{2,\ldots,\alpha} _{n\right)} is a multiindex, style \left\vert \alpha\right\vert = style \alpha _{1+\alpha} _{2+ … +\alpha} _{n} and style D ^{\alpha} denotes the derivative style \alpha _{1} times with respect to style x _{1} , style \alpha _{2} times with respect to style x _{2} , and so on. The Sobolev seminorm on style W ^{m(\Omega)} _{p} consists of the style L ^{p} norms of the highest order derivatives,\left\vert u\right\vert ^{W ^{m(\Omega)} _{p}} = \left( \sum _{\left\vert} \left\Vert D ^{\alpha} u\right\Vert ^{p\right)} _{L ^{p(\Omega)}} ^{1/p} if 1 \leq p< inf ty

and

\left\vert u\right\vert ^{W ^{m} _{inf ty} (\Omega)} =max _{\left\vert} \left\Vert D ^{\alpha} u\right\Vert ^{L ^{infty(\Omega)}} style P _{k} is the space of all polynomials of order up to style k on style R ^{n} . Note that style D ^{\alpha} v =0 for all style v\in P _{m-1} and style \left\vert \alpha\right\vert = m , so style \left\vert u + v\right\vert ^{W ^{m(\Omega)} _{p}} has the same value for any style v\in P _{m-1} .

Lemma (Bramble and Hilbert) Under additional assumptions on the domain

style \Omega , specified below, there exists a constant style C = C\left( m,\Omega\right) independent of style p and style u such that for any style u\in W ^{m(\Omega)} _{p} there exists a polynomial style v\in P _{m-1} such that for all style k =0 ,\ldots,m, \left\vert u - v\right\vert ^{W ^{k(\Omega)} _{p}} \leq Cd ^{m-k} \left\vert u\right\vert ^{W ^{m(\Omega)} _{p}} .

The original result

The lemma was proved by Bramble and Hilbert ^[1] under the assumption that

style \Omega satisfies the strong cone property; that is, there exists a finite open covering style \left\{ O _{i} \right\} of style \partial\Omega and corresponding cones style \{C _{i} \} with vertices at the origin such that style x + C _{i} is contained in style \Omega for any style x style \in\Omega\cap O _{i} .

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

style W ^{m} _{p} (\Omega)/P _{m-1} is equivalent to the style W ^{m} _{p} (\Omega) seminorm. The style W ^{m} _{p} (\Omega) norm is not the usual one but the terms are scaled with style 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

style \Omega cannot be determined from the proof.

A constructive form

An alternative result was given by Dupont and Scott ^[2] under the assumption that the domain

style \Omega is star-shaped; that is, there exists a ball style B such that for any style x\in\Omega , the closed convex hull of style \left\{ x\right\} \cup B is a subset of style \Omega . Suppose that style \rho _{max} is the supremum of the diameters of such balls. The ratio style \gamma = d/\rho _{max} is called the chunkiness of style \Omega .

Then the lemma holds with the constant

style C = C\left( m,n,\gamma\right) , that is, the constant depends on the domain style \Omega only through its chunkiness style \gamma and the dimension of the space style n . In addition, v can be chosen as v = Q ^{m} u , where style Q ^{m} u is the averaged Taylor polynomial, defined asQ ^{m} u = \int _{B} T ^{mu\left(} _{y} x\right) \psi\left( y\right) dx,

where

T ^{m} _{y} u\left( x\right) = \sum\limits ^{m -1} _{k =0} \sum\limits _{\left\vert} (1)/(\alpha!) D ^{\alpha} u\left( y\right) \left( x - y\right) ^{\alpha}

is the Taylor polynomial of degree at most

style m -1 of style u centered at style y evaluated at style x , and style \psi\geq 0 is a function that has derivatives of all orders, equals to zero outside of style B , and such that\int _{B\psi} dx =1 .

Such function

style \psi 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

style \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.^[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

style \ell is a continuous linear functional on style W ^{m} _{p} (\Omega) and style \left\Vert \ell\right\Vert ^{W ^{m} _{p} (\Omega ) ^{^{\prime}}} its dual norm. Suppose that style \ell\left( v\right) =0 for all style v\in P _{m-1} . Then there exists a constant style C = C\left( \Omega\right) such that\left\vert \ell\left( u\right) \right\vert \leq C\left\Vert \ell\right\Vert ^{W ^{m} _{p} (\Omega) ^{^{\prime}}} \left\vert u\right\vert ^{W ^{m} _{p} (\Omega)} .

External links

• • https://arxiv.org/abs/0710.5148 – Jan Mandel: The Bramble–Hilbert Lemma

Notes and References

1. 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.
2. Todd Dupont and Ridgway Scott. Polynomial approximation of functions in Sobolev spaces. Math. Comp., 34(150):441–463, 1980.
3. [Susanne Brenner|Susanne C. Brenner]
4. [Philippe G. Ciarlet]