Symmetric and nonsymmetric Nitsche boundary forms

Assume Countable Choice. Let $d\ge 2$, let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a nonempty bounded connected polyhedral open set, and let ${\mathcal{T}}_h$ be a finite matching affine simplicial mesh of $\mathrm{\Omega }$ as in Matching affine simplicial meshes and shape regularity. Fix a polynomial degree $k\ge 1$. Let $V_h$ be the real scalar conforming polynomial space of Broken and conforming polynomial spaces on a matching mesh, viewed as a subspace of $H^1(\mathrm{\Omega })$ by Continuous matching piecewise polynomials belong to W1p, and let $Y_h:=V_h\cap H_0^1(\mathrm{\Omega })$, where $H_0^1$ is the closure convention of Zero-boundary Sobolev space as a norm closure; Zero facet traces of functions in H1_0 on an affine mesh proves its facet-trace consequence. Boundary facets form ${\mathcal{F}}_h^{\mathrm{\partial }}$. If $F\in {\mathcal{F}}_h^{\mathrm{\partial }}$, let $K_F$ be its unique incident cell, $n_F$ its outward unit normal, $\mathrm{\mid }F\mathrm{\mid }$ its $(d-1)$-dimensional measure, and set $h_F:=\mathrm{\mid }{K}_F\mathrm{\mid }\mathrm{/}\mathrm{\mid }F\mathrm{\mid }>0$. For functions with a defined boundary normal trace write ${\mathrm{\partial }}_{n}v:=\mathrm{\nabla }v\cdot n_F$ on $F$.

Let ${\mathcal{X}}_h$ consist of $v\in H^1(\mathrm{\Omega })$ such that for every cell $K$ there is $s_K>1\mathrm{/}2$ with $v{\mathrm{\mid }}_K\in H^{1+s_K}(K)$. By Fractional Bessel-potential traces on simplex facets, $v$ has an $L^2(F)$ trace on each facet, and each component of $\mathrm{\nabla }(v{\mathrm{\mid }}_K)$ is in $H^{s_K}(K)$, hence has an $L^2(F)$ trace by the same lemma; the derivative mapping follows from $\mathrm{\mid }{\xi }_j\mathrm{\mid }⟨\xi {⟩}^{s_K}\le ⟨\xi {⟩}^{1+s_K}$ in the Bessel-potential norm. Thus ${\mathrm{\partial }}_{n}v$ and all displayed integrals are defined. In particular, $V_h\subseteq {\mathcal{X}}_h$ and the exact/adjoint functions in the later regularity hypotheses lie in ${\mathcal{X}}_h$. For a dimensionless penalty coefficient ${\gamma }_0\ge 0$ and symmetry parameter $\theta \in \{0,1\}$, define the same bilinear formula on ${\mathcal{X}}_h\times {\mathcal{X}}_h$: $a_h^{\theta }(v,w):={\int }_{\mathrm{\Omega }}\mathrm{\nabla }v\cdot \mathrm{\nabla }wdx-{\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{\int }_F({\mathrm{\partial }}_{n}v)wdS-\theta {\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{\int }_{F}v({\mathrm{\partial }}_{n}w)dS+{\gamma }_0{\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{h}_F^{-1}{\int }_{F}vwdS\mathrm{.}$ For $f\in L^2(\mathrm{\Omega })$ and boundary data $g$ with the displayed integrals finite, define ${\mathrm{\ell }}_h^{\theta }(w_h):={\int }_{\mathrm{\Omega }}f{w}_{h}dx-\theta {\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{\int }_{F}g({\mathrm{\partial }}_n{w}_h)dS+{\gamma }_0{\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{h}_F^{-1}{\int }_{F}g{w}_{h}dS\mathrm{.}$ The nonsymmetric Nitsche form is $a_h^0$; it has the consistency flux in the trial slot only. The symmetric Nitsche form is $a_h^1$; it has both transposed consistency fluxes. The discrete weak-boundary problem is the standard Nitsche method when ${\gamma }_0>0$: it seeks $u_h\in V_h$ such that $a_h^{\theta }(u_h,w_h)={\mathrm{\ell }}_h^{\theta }(w_h)$ for every $w_h\in V_h$. This extension makes expressions such as $a_h^{\theta }(u,w_h)$ and $a_h^{\theta }(e,\zeta -z_h)$ well-defined when the exact and adjoint functions have the stated cellwise regularity. The boundary penalty is ${\gamma }_0\mathrm{/}{h}_F$; $h_F$ is the cell-face height divided by $d$, not the face diameter. The zero value ${\gamma }_0=0$ is used only for the penalty-free algebraic comparison.

The energy seminorm and the comparison seminorm used below are $\mathrm{\parallel }v{\mathrm{\parallel }}_{E,h}^2:=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2+{\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{h}_F^{-1}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(F)}^2,\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\#},h}^2:=\mathrm{\parallel }v{\mathrm{\parallel }}_{E,h}^2+{\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{h}_F\mathrm{\parallel }{\mathrm{\partial }}_{n}v{\mathrm{\parallel }}_{L^2(F)}^2\mathrm{.}$ These expressions are used only on functions for which their volume and facet traces are defined. The parameter $\theta$ fixes the sign convention; no convention based on a reciprocal small stabilization parameter is implicit.