Bramble–Hilbert lemma

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 $h^{m-k}$.

Going deeper

For a student

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.