Finite element correction and strong convergence of Whitney cochain projections

On each indicated finite element space the Neumann series $J_h={\sum }_{j=0}^{\mathrm{\infty }}(I-A_h{)}^j$ converges in operator norm, has norm at most two, and telescopes to both inverse identities. Therefore ${\pi }_h$ has norm at most $2C{ϵ}^{-3\mathrm{/}2}$ and equals the identity on its target. Its range is precisely that target, so ${\pi }_h^2={\pi }_h$. For $k=0,1,2$, the restriction of commutation to the subcomplex gives $d{A}_h^k=A_h^{k+1}d$; applying this identity to $J_h^k{v}_h$ and multiplying by $J_h^{k+1}$ gives $d{J}_h^k{v}_h=J_h^{k+1}d{v}_h$. Consequently $d{\pi }_h^{k}u={\pi }_h^{k+1}du$ on the stated full or relative domain in those degrees. In degree three, $d=0$, so the terminal commutation identity is automatic and needs no degree-four operator. The two $L^2$ bounds give the graph-norm bound directly.

The projection identity yields $\mathrm{\parallel }u-{\pi }_{h}u{\mathrm{\parallel }}_2\le (1+\mathrm{\parallel }{\pi }_h\mathrm{\parallel }){\inf }_{v_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_2$. For a globally smooth form $u$ on a neighborhood of $\bar{\mathrm{\Omega }}$, canonical Whitney interpolation converges in $L^2$: subtract a constant-coefficient form on each cell, which is reproduced in every degree; the remaining coefficient has supremum at most $C{h}_K{\sup }_K\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$. Its simplex moments are at most $C{h}_K^{k+1}{\sup }_K\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$, so the scaled moment norm gives $\mathrm{\parallel }u-I_{h}u{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}2}{\sup }_K\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$. Summing proves convergence. Relative spaces have the same approximation property for compactly supported smooth inputs, whose boundary moments vanish once the mesh is sufficiently fine. By [F1], smooth restrictions are dense in full $L^2$, and compactly supported smooth forms are dense in relative $L^2$, which is the same ambient $L^2$ space. A three-term approximation and the uniform projection bound prove strong $L^2$ convergence for every $u$ in either case.

If $\mathcal{C}$ is compact in $L^2$, choose a finite $\delta$-net $\{u_1,\dots ,u_N\}$ in it. For $u$ within $\delta$ of $u_j$, $\mathrm{\parallel }(I-{\pi }_h)u{\mathrm{\parallel }}_2\le (1+C)\delta +\mathrm{\parallel }(I-{\pi }_h)u_j{\mathrm{\parallel }}_2$. The finite maximum tends to zero by step 2.1, and then $\delta \downarrow 0$ proves uniform convergence on $\mathcal{C}$. For $u$ in a graph domain of degree $k<3$, commutation and strong convergence in degrees $k,k+1$ give $\mathrm{\parallel }u-{\pi }_{h}u{\mathrm{\parallel }}_2+\mathrm{\parallel }du-d{\pi }_{h}u{\mathrm{\parallel }}_2\to 0$. In degree three the graph norm is the $L^2$ norm, so step 2.1 gives graph convergence.