The Bramble-Hilbert lemma for convex domains

The Bramble-Hilbert lemma is a fundamental result on multivariate polynomial approximation. It is frequently applied in the analysis of finite elements methods (FEM) used for numerical solutions of PDEs. However, this classical estimate depends on the geometry of the domain and may "blow up" for simple examples such as a sequence of triangles of equivalent diameter that become thinner and thinner. Thus, in FEM applications one usually requires that the mesh has "quasi-uniform" geometry. This assumption is perhaps too restrictive when one tries to obtain estimates of nonlinear approximation methods that use piecewise polynomials. Our main result that improves upon this point is the following. Let Ω ⊂ ℝⁿ be a bounded convex domain and let g ∈ W_p^m(Ω), m ∈ ℕ, 1 ≤ p ≤ ∞, where W_p^m(Ω) is the Sobolev space. Then there exists a polynomial P of total degree m - 1 for which |g - P|_k,P ≤ C(n, m)(diam Ω)^m-k|g| _m,p, k = 0, 1,..., m, where|·|_k,p := Σ _|α|=k ∥D^α ∥L_p(Ω) is the Sobolev seminorm of order k. As a consequence we get that for f ∈ L_p(Ω), E_m-1(f, Ω)_p ≈ K _m(f, (diam Ω)^m)_p, where E_m-1 (f, Ω)_p := inf_p∈m-1∥ f-P∥ _Lp(Ω) is the error of polynomial approximation of degree m - 1 and K_m( , )_p is the K-functional associated with the pair (L_p(Ω), W_p^m(Ω)), and where the constants of equivalence depend only on m and n. For the case of convex domains (elements) this extends a recent result for p = 2, and for m = 1 and 2 < p ≤ ∞. This also improves previous results where the constant in the estimate further depends on the geometry of the domain, or where there is a constraint p > n(≥ 2).