Nonsymmetric Nitsche L2 estimate with its boundary consistency defect

Given: Countable Choice; the nonsymmetric method and its energy orthogonality; $e=u-u_h^0$; the zero-trace adjoint solution; and [A1], its stated dual regularity.

The energy theorem gives $a_h^0(e,w_h)=0$ for every $w_h\in V_h$; its weighted Cauchy bound at $\theta =0$ has constant $2+{\gamma }_0$, denoted $M_0$ here (Nitsche consistency and energy quasi-optimality).

$H_0^1(\mathrm{\Omega })$ is formed from $W^{1,2}(\mathrm{\Omega })$ using the zero-trace closure convention (Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure), and with its usual norm is a Hilbert space (The Sobolev space $H^1$ is a Hilbert space). The adjoint Poisson form is coercive in that norm by the Poincare inequality, and the adjoint variational problem is well-posed under Countable Choice (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, Well-posedness of the adjoint variational problem).

$H^s(K)$ is the restriction of $H^s({\mathbb{R}}^d)$ with the quotient Bessel-potential norm, and the weighted Fourier norm maps $H^{1+s}({\mathbb{R}}^d)$ continuously into $H^s({\mathbb{R}}^d)$ under each first derivative (Restriction Bessel-potential spaces on polyhedral domains, Real-order Bessel-potential completion H^s, Real-order H^s as weighted Fourier distributions).

For $s>1\mathrm{/}2$, a function in $H^s(K)$ has a unique bounded $L^2$ trace on each affine simplex facet (Fractional Bessel-potential traces on simplex facets).

Every real function in $H_0^1(\mathrm{\Omega })$ has zero $L^2$ trace on each boundary facet (Zero facet traces of functions in H1_0 on an affine mesh).

If $v\in H^{1+s}(\mathrm{\Omega })$, $-\mathrm{\Delta }v=g\in L^2(\mathrm{\Omega })$ distributionally, and the cellwise normal derivatives have the stated fractional traces, the weak Green supplier gives the global Green identity with interior flux cancellation (Fractional weak Green identity and interior flux cancellation).

The zero-boundary fractional comparison interpolation theorem supplies $z_h=I_{h,0}{\zeta }_e$ and the full comparison-norm bound in the Statement (Zero-boundary fractional interpolation in the Nitsche comparison norm).

The fixed-domain weighted boundary flux trace theorem gives the stated estimate for every ${\zeta }_e\in H^{1+s}(\mathrm{\Omega })$ (Mesh-uniform fractional boundary flux trace).

The adjoint solution satisfies $\mathrm{\parallel }{\zeta }_e{\mathrm{\parallel }}_{H^{1+s}(\mathrm{\Omega })}\le C_R\mathrm{\parallel }e{\mathrm{\parallel }}_2$ for some fixed $C_R$ and $s\in (1\mathrm{/}2,1]$. This dual elliptic regularity is an explicit PDE hypothesis, not a consequence of the variational adjoint equation.