Lecture 3 — Polynomial approximation and a priori error estimates

The main arguments developed in this lecture are

Objectives and prerequisites¶

By the end of the lecture, you should be able to prove the equivalence between a Sobolev seminorm and a quotient norm modulo polynomials, estimate an operator that annihilates polynomials, transfer reference-cell estimates to a physical mesh, and derive an energy-norm error bound for a conforming Galerkin method.

We use Sobolev spaces, affine scaling and shape regularity from Lecture 2, and Theorem 2 from Lecture 1. We recall the precise compactness and embedding results used below; their proofs belong to the theory of Sobolev spaces. The construction and unisolvence of general Lagrange elements will be developed in Lecture 4. Here their required properties are stated explicitly.

Throughout, $m\geq1$ is an integer, $1\leq p<\infty$, and $\mathbb P^{m-1}(D)$ denotes polynomials of total degree at most $m-1$ restricted to $D$. Constants may depend on $m,p,d$ and the fixed domain; mesh-independent estimates will state their geometric assumptions.

1. The kernel of the highest-order seminorm¶

On a bounded, connected Lipschitz domain $D\subset\mathbb R^d$, the seminorm

vanishes precisely on $\mathbb P^{m-1}(D)$.

To justify the nontrivial implication, first recall that a distribution with zero gradient on a connected open set is constant. Locally this follows by mollification on balls: its mollifications have zero gradient and are constant on smaller balls. Passing to the distributional limit and using overlapping balls gives the same constant throughout the connected set.

If all order-$m$ derivatives of $v$ vanish, every derivative $D^\beta v$ with $|\beta|=m-1$ is therefore a constant $c_\beta$. The polynomial

has exactly those order-$(m-1)$ derivatives. Subtract it and apply induction to $v-q_{m-1}$. This gives a polynomial of degree at most $m-1$ representing $v$ almost everywhere. The case $m=1$ is the zero-gradient assertion.

Connectedness matters: on a disconnected domain, the kernel contains functions that are independently polynomial on each connected component.

2. The Deny–Lions lemma¶

Proof. Set $X=W^{m,p}(D)$ and $Q=\mathbb P^{m-1}(D)$. The first inequality follows from $|v-q|_{W^{m,p}}=|v|_{W^{m,p}}$. For the other inequality, choose a basis $q_1,\ldots,q_N\in Q$ and its dual basis $q^1,\ldots,q^N\in Q'$, characterized by $q^i(q_j)=\delta_{ij}$. Here $Q$ carries the norm inherited from $X$; finite dimensionality makes every $q^i$ continuous.

The inclusion $j:Q\hookrightarrow X$ induces the bounded restriction map

Hahn–Banach makes $j'$ surjective: every functional on $Q$ admits a continuous extension to $X$. There is no canonical inclusion of $Q'$ into $X'$ from $j$ alone, because these extensions are generally not unique. Choose extensions $\lambda_i\in X'$ of the dual basis functionals, so that $j'\lambda_i=q^i$ and $\lambda_i(q_j)=\delta_{ij}$. These choices define a bounded linear map

Thus $E$ identifies $Q'$ with a subspace of $X'$ after a choice of extensions; this identification is noncanonical. Define the bounded projection

We prove

If this failed, normalization would give a sequence $v_n$ with $\|v_n\|_X=1$ and both $|v_n|_{W^{m,p}}\to0$ and $\lambda_i(v_n)\to0$. The compact embedding $W^{m,p}(D)\hookrightarrow W^{m-1,p}(D)$ provides a subsequence converging strongly in $W^{m-1,p}(D)$. Moreover,

Consequently, this subsequence is Cauchy in $X$ and converges strongly in $X$ to some $v$. Continuity gives $\|v\|_X=1$, $|v|_{W^{m,p}}=0$ and $\lambda_i(v)=0$ for every $i$. The kernel characterization implies $v\in Q$; its zero coordinates imply $v=0$, a contradiction.

Apply (8) to $v-Rv$. All its coordinates vanish, and its order-$m$ derivatives equal those of $v$. Hence

This proves the lemma. The compact embedding used here holds for the stated bounded Lipschitz domains and finite $p$, including $p=1$; no weak subsequence argument or reflexivity is needed. See Brenner & Scott (2010) for the Sobolev-space framework.

The projection $R$ is an auxiliary device for this proof. It does not prescribe nodal values or ensure matching traces across a mesh.

3. Bramble–Hilbert for operators that annihilate polynomials¶

Proof. For every $q\in\mathbb P^{m-1}(D)$, linearity gives $\tau v=\tau(v-q)$. Bound this by $\|\tau\|\|v-q\|_{W^{m,p}}$, take the infimum over $q$, and apply Deny–Lions.

In particular, if a bounded linear operator $I$ reproduces these polynomials, then $\tau=\mathrm{Id}-I$ satisfies the hypothesis. Both boundedness on the chosen Sobolev space and polynomial reproduction must be checked. Reproduction alone does not provide an error estimate.

4. Reference-cell interpolation¶

Fix a reference simplex $\widehat T$ and a polynomial degree $k\geq1$. For an integer $1\leq m\leq k+1$, assume that a linear reference operator $\widehat I_k$ has the following properties:

1. $\widehat I_k:W^{m,p}(\widehat T)\to\mathbb P^k(\widehat T)$ is bounded;

2. $\widehat I_k q=q$ for every $q\in\mathbb P^k(\widehat T)$.

Since polynomial norms are equivalent on this fixed finite-dimensional space, $\widehat I_k$ is bounded into $W^{s,p}(\widehat T)$ for every integer $0\leq s\leq m$. Bramble–Hilbert applied to $\mathrm{Id}-\widehat I_k$ therefore gives

For nodal interpolation, these assumptions hold if the nodes are unisolvent for $\mathbb P^k$ and $mp>d$. Indeed, Sobolev embedding gives a continuous representative and bounds every nodal evaluation by $C\|\widehat v\|_{W^{m,p}}$. Writing $\widehat I_k\widehat v=\sum_i\widehat v(\widehat a_i)\widehat\varphi_i$ then proves boundedness; unisolvence proves reproduction.

We will establish unisolvence for the standard Lagrange nodes in Lecture 4. The condition $mp>d$ is a sufficient condition used throughout the nodal argument here. In particular, arbitrary $H^1$ functions in two dimensions do not have well-defined nodal values. Interpolants for low-regularity functions will be introduced later.

5. Local estimates and shape regularity¶

Let $\mathcal T_h$ be a conforming simplicial mesh of a polygonal or polyhedral domain, with affine maps $F_T(\widehat x)=B_T\widehat x+b_T$. Set $h_T=\operatorname{diam}(T)$, let $\rho_T$ be its inradius, and let $J_T=|\det B_T|$. Define the transported operator by

This definition, and not merely the fact that $I_Tv$ is polynomial, ensures that interpolation commutes with pullback. For $e_T=v-I_Tv$, reference-cell scaling and (16) give

Here the order-zero seminorm means the $L^p$ norm. The determinant factors cancel. Since $\|B_T\|\leq Ch_T$ and $\|B_T^{-1}\|\leq C\rho_T^{-1}$, this yields $|e_T|_{W^{s,p}(T)}\leq Ch_T^m\rho_T^{-s}|v|_{W^{m,p}(T)}$. For a shape-regular family, $h_T/\rho_T\leq\sigma^{-1}$ uniformly, so

The constant depends on $k,m,p,d$, the reference element, its interpolation operator and $\sigma$, but not on $T$ or its size. We keep $k$ fixed; this argument gives no uniform estimate as $k\to\infty$.

For $h_T\leq1$, applying this estimate at every derivative order $j\leq s$ also gives $\|v-I_Tv\|_{W^{s,p}(T)}\leq Ch_T^{m-s}|v|_{W^{m,p}(T)}$. This follows from $h_T^{m-j}\leq h_T^{m-s}$. It does not require lower-order derivatives of $v$ to be controlled by its highest-order seminorm.

6. From local approximation to a conforming function¶

Local estimates initially control a broken norm. To use Céa’s lemma we also need a global interpolant $I_hv\in V_h\subset H_0^1(\Omega)$. Assume that the local operators use matching Lagrange nodes on common faces, that those face nodes are unisolvent for the trace polynomial space, and that boundary nodes prescribe homogeneous Dirichlet data. For a continuous $v$ with zero boundary values, the resulting $I_hv$ has matching traces and zero trace on $\partial\Omega$.

For $P_1$ triangles this can be checked immediately: the affine interpolant is determined by the three vertex values; two neighboring triangles give linear edge traces with the same two endpoint values. Those traces agree on the entire edge. A boundary edge has two zero endpoint values and hence zero trace. For higher degrees we use the analogous face-unisolvence property, whose construction is deferred to Lecture 4.

Matching traces imply $I_hv\in H^1(\Omega)$: integration by parts cell by cell against a smooth compactly supported test function cancels the interior face terms, leaving the piecewise gradient as the distributional gradient. It lies in $L^2$, and zero boundary trace gives $I_hv\in H_0^1$. Independent polynomial projections on each cell generally lack this property.

7. The energy-norm error bound¶

Proof. Here $m>d/2$, so nodal evaluation is bounded and $u$ has a continuous representative. Its zero trace gives zero boundary values. Consequently, $I_hu\in V_h$. Céa and the local estimate with $s=1$, $p=2$ give

Since $h_T\leq h$ and the cells partition $\Omega$, the global bound follows. Poincaré’s inequality gives the corresponding full $H^1$-norm bound. No comparison between the smallest and largest cell sizes was used: shape regularity suffices; quasi-uniformity is unnecessary for this estimate.

Poisson and solution regularity¶

For $a(w,v)=\int_\Omega\nabla w\cdot\nabla v$, the energy norm chosen above gives $M=\alpha=1$. Thus, for $P_1$ triangles and $u\in H^2\cap H_0^1$,

If $\Omega$ is a bounded convex polygon and $f\in L^2(\Omega)$, elliptic regularity gives $u\in H^2(\Omega)$ and $\|u\|_{H^2}\leq C_\Omega\|f\|_{L^2}$. Invoking this separate PDE result then yields $|u-u_h|_{H^1}\leq CC_\Omega h\|f\|_{L^2}$; see Brenner & Scott (2010) and Ern & Guermond (2004) for regularity and finite element error estimates. Convexity does not provide arbitrary higher regularity at polygonal corners.

For degree $k$, the rate is $h^{m-1}$ at the available integer regularity $2\leq m\leq k+1$. Increasing $k$ improves the proven rate only when the solution has the additional regularity. A general Lipschitz domain with $f\in L^2$ need not give $u\in H^2$; the present nodal proof does not cover all low-regularity solutions. Fractional regularity and appropriate quasi-interpolants require further analysis.

The estimate $\|u-I_hu\|_{L^2}\leq Ch^m|u|_{H^m}$ concerns interpolation. Céa in the energy norm does not imply the same $L^2$ rate for $u-u_h$. That additional Galerkin estimate will use duality in Lecture 5.

8. Exercises¶

1. Poincaré modulo constants. For $m=1$, use the bounded functional $\lambda(v)=|D|^{-1}\int_D v$ in the Deny–Lions proof and deduce $\|v-\lambda(v)\|_{W^{1,p}(D)}\leq C|v|_{W^{1,p}(D)}$.

2. Local-to-global estimate. For $P_1$ triangles, derive $|u-u_h|_{H^1}^2\leq C\sum_T h_T^2|u|_{H^2(T)}^2$. Identify the assumptions that guarantee $I_hu\in H_0^1$ and explain why replacing each local interpolant with an independent $L^2$ projection does not justify this proof.

3. Scaling check, independent study. On $(0,h)$, interpolate $v(x)=x^2$ at the endpoints. Compute exactly $\|v-I_1v\|_{L^2}=h^{5/2}/\sqrt{30}$ and $|v-I_1v|_{H^1}=h^{3/2}/\sqrt3$, and compare them with $h^2|v|_{H^2}$ and $h|v|_{H^2}$. Here $|v|_{H^2}=2\sqrt h$.

Further reading¶

Polynomial approximation, reference-element interpolation and conforming error estimates are treated in Ciarlet (1978), Brenner & Scott (2010) and Ern & Guermond (2004). The central proofs above are complete; the compact Sobolev embedding, the embedding into continuous functions and the convex-polygon elliptic regularity result are invoked with explicit assumptions.