Symmetric Nitsche L2 estimate under dual regularity and Nitsche approximation

Given: Countable Choice; the symmetric weak-boundary problem and its energy orthogonality; $e=u-u_h^1$; the zero-trace adjoint solution ${\zeta }_e$; its stated dual regularity; and the remaining primal approximation hypothesis only for the final conditional branch.

The energy theorem gives $a_h^1(e,w_h)=0$ for all $w_h\in V_h$ (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 variational equation is well-posed for a bounded coercive form on a Hilbert space under Countable Choice (Well-posedness of the adjoint variational problem). On bounded $\mathrm{\Omega }$, the Poincare inequality controls the $L^2$ part of the $W^{1,2}$ norm of a zero-trace function by its first-derivative part (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

$H^s(K)$ 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 and interior flux cancellation for every $w\in H^1(\mathrm{\Omega })$ (Fractional weak Green identity and interior flux cancellation).

The zero-boundary fractional comparison interpolation theorem supplies $z_h=I_{h,0}{\zeta }_e\in Y_h$ and the displayed $C_D{h}^s{\mathrm{\ell }}_{\mathrm{\Omega }}^{1-s}$ estimate (Zero-boundary fractional interpolation in the Nitsche comparison norm).

For the nonempty domain, $C_{\mathrm{\partial }}\ge 1$; the forms have one volume term, two consistency flux terms, and one penalty term with weights matching $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\#},h}$ (Global boundary-flux estimate on a matching mesh, Symmetric and nonsymmetric Nitsche boundary forms).

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.

The primal solution satisfies the energy theorem's comparison-norm approximation hypothesis, so its conditional conclusion gives $\mathrm{\parallel }e{\mathrm{\parallel }}_{\mathrm{\#},h}\le C_{\theta }{C}_I{\left({\sum }_{K\in {\mathcal{T}}_h}{h}_K^{2t}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{1+t}(K)}^2\right)}^{1\mathrm{/}2}$ (Nitsche consistency and energy quasi-optimality).