Conforming FEM families with stable L2 projectors

Let $(V,H,J)$ be the Banach evolution triple of Banach evolution triples and weak time derivatives. An admissible stable conforming FEM sequence consists of finite-dimensional subspaces $V_k\subset V$ and their $H$-orthogonal projections ${\mathrm{\Pi }}_k:H\to V_k$ such that:

1. ${\mathrm{\Pi }}_k(V)\subset V$, ${\mathrm{\Pi }}_{k}v=v$ for $v\in V_k$, and $(w_k,z-{\mathrm{\Pi }}_{k}z{)}_H=0$ for every $w_k\in V_k$, $z\in H$;
2. ${\sup }_k\mathrm{\parallel }{\mathrm{\Pi }}_k{\mathrm{\parallel }}_{V\to V}\le C_{\mathrm{\Pi }}<\mathrm{\infty }$ and ${\mathrm{\Pi }}_{k}v\to v$ in $V$ for each $v\in V$;
3. the associated time partitions of $[0,T]$ have maximum step ${\delta }_k\to 0$.

The convergence in clause 2 and density of $V$ in $H$ imply ${\mathrm{\Pi }}_{k}z\to z$ in $H$ for every $z\in H$: use $\mathrm{\parallel }{\mathrm{\Pi }}_k(z-v){\mathrm{\parallel }}_H\le \mathrm{\parallel }z-v{\mathrm{\parallel }}_H$, ${\mathrm{\Pi }}_{k}v\to v$ in $H$, and approximate $z$ by $v\in V$. Thus the range spaces approximate every $H$ vector.

For a fixed $k$, backward Euler with averages $f_k^n$ uses unknowns $U_k^n\in V_k$ and equations ${(\frac{U_k^n-U_k^{n-1}}{{\tau }_n},v_k)}_H+⟨A(U_k^n),v_k⟩=⟨f_k^n,v_k⟩(v_k\in V_k)\mathrm{.}$ Its piecewise-constant reconstruction is ${\bar{U}}_k(t)=U_k^n$ on $(t_{n-1},t_n]$; its affine reconstruction joins $U_k^{n-1}$ to $U_k^n$.

The uniform $V$-stability of the $H$-orthogonal projections is an explicit method hypothesis; conformity alone does not imply it. The same-page L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes proves this interface for fixed-degree globally quasi-uniform shape-regular simplex meshes on bounded Lipschitz polyhedral domains in $d\ge 2$ and for the directly defined interval meshes in $d=1$. The abstract definition remains applicable to other conforming families when its projector clauses are independently verified.