Interpolation estimates and Bramble–Hilbert

On a fixed bounded connected reference domain with suitable Lipschitz geometry, the Bramble–Hilbert or Deny–Lions approximation result bounds the distance to polynomials of degree at most m minus one by the order-m Sobolev seminorm. For uniformly scaled geometric families one needs uniform control of the reference-domain constants; a fixed reference simplex supplies it. More general varying domains require geometric assumptions such as controlled star-shapedness rather than a claim that one constant works on every possible thin or disconnected region.

Suppose an interpolation operator reproduces those polynomials and is bounded from the chosen Sobolev source space into the desired target space. Subtract an approximating polynomial before applying the operator error. Reproduction eliminates that polynomial exactly, and boundedness controls the remaining function. This converts a full Sobolev norm estimate into one involving only the highest derivatives. An unbounded point-evaluation functional cannot be rescued merely by observing that a formal polynomial formula reproduces linear functions.

For continuous piecewise linear nodal interpolation from H² on a simplex, point evaluation is bounded when two exceeds half the spatial dimension. Thus the familiar H²-to-nodal-P1 argument applies directly in dimensions one, two, and three. An arbitrary H¹ function in two or three dimensions does not carry well-defined point values at every mesh vertex. Treating a representative’s arbitrary pointwise choices as interpolation data would make the construction depend on choices invisible to the Sobolev norm.

Quasi-interpolation operators based on local averages can avoid this obstacle and are useful for lower-regularity solutions and a posteriori analysis. They have their own locality, boundary, and stability properties to verify. For nodal P1 interpolation with u in H² and dimensions at most three, affine scaling on a shape-regular mesh yields the local L² and gradient estimates below. Global estimates follow by summing squared local errors, retaining local cell sizes on a graded mesh.

The estimates assume the stated derivatives exist. They do not prove that every weak solution of a polygonal Dirichlet problem lies in H². Reentrant corners, coefficient interfaces, and rough data can reduce regularity. In that case a convergence proof must use the available weaker approximation result or a mesh adapted to the actual singularity, rather than inserting an unjustified H² norm into an error bound. Polynomial degree alone cannot restore smoothness absent from the solution.

Similarly, local estimates with mesh-independent constants do not predict an exact error ratio for every particular function. An affine function has zero P1 interpolation error for every cell size, while a quadratic exposes the leading power clearly. Exact calculations on simple polynomials are useful audits of the scaling powers and derivative factors. They illustrate the theorem’s mechanism without proving the general estimate or establishing that the same global rate holds for a PDE solution of unknown regularity.