A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems

Abstract

We present a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear elliptic problem posed on a piecewise curved, non polygonal domain. We approximate $\Omega$ by a polygonal subdomain $\Omega _h$ and propose an HDG discretization, which is shown to be optimal under mild assumptions related to the non-linear source term and the distance between the boundaries of the polygonal subdomain $\Omega _h$ and the true domain $\Omega$. Moreover, a local non-linear post-processing of the scalar unknown is proposed and shown to provide an additional order of convergence. A reliable and locally efficient a posteriori error estimator that takes into account the error in the approximation of the boundary data of $\Omega _h$ is also provided.

Appendices

HDG projection

In order to make this manuscript self-contained, in this section we provide previous results that will help us to analyze our discrete scheme. First of all we recall the HDG projection operators introduced by [7]. Given constants $l_u, l_{\varvec{q}} \in [0,k]$ and a pair of functions $({\varvec{q}},u) \in H^{1+l_q}(T) \times H^{1+l_u}(T)$, we denote by $\varvec{\Pi }(\varvec{q},u):=(\varvec{\Pi }_{\mathrm v}\varvec{q},\Pi _{\mathrm w} u)$ the projection over $\varvec{V}_h\times W_h$ defined as the unique element-wise solutions of

$$\begin{aligned} (\varvec{\Pi }_{\mathrm v}\varvec{q}, \varvec{v})_T&= (\varvec{q}, \varvec{v})_T&\forall \ \varvec{v} \in [\mathbb {P}_{k-1}(T)]^d, \end{aligned}$$

(A.1a)

$$\begin{aligned} (\Pi _{\mathrm w} u, w)_T&= (u,w)_T&\forall \ w\in \mathbb {P}_{k-1}(T), \end{aligned}$$

(A.1b)

$$\begin{aligned} \left\langle \varvec{\Pi }_{\mathrm v}\varvec{q}\cdot \varvec{n} + \tau \Pi _{\mathrm w} u, \mu \right\rangle _{F}&= \left\langle \varvec{q} \cdot \varvec{n} + \tau u, \mu \right\rangle _F&\forall \ \mu \in \mathbb {P}_k(F), \end{aligned}$$

(A.1c)

for every element $T\in \mathcal {T}_h$, and $F\in \partial T$. The $L^2$ projection into $M_h$ will be denoted as $P_M$. If the stabilization function is chosen so that $\tau _T^{\max } := \max \tau |_{\partial T}>0$, then by [7] there is a constant $C>0$ independent of T and $\tau$ such that

$$\begin{aligned} \Vert \varvec{\Pi }_{\mathrm v}\varvec{q} - \varvec{q}\Vert _T&\le C h_T^{l_{\varvec{q}}+1} |\varvec{q}|_{\varvec{H}^{l_{\varvec{q}}+1}(T)} + C h_T^{l_u+1} \tau _T^* |u|_{H^{l_u+1}(T)}, \end{aligned}$$

(A.2a)

$$\begin{aligned} \Vert \Pi _{\mathrm w} u - u\Vert _T&\le C h_T^{l_u+1} |u|_{H^{l_u+1}(T)} + C \dfrac{h_T^{l_{\varvec{q}}+1}}{\tau _T^{\max }} |\nabla \cdot \varvec{q}|_{H^{l_{\varvec{q}}}(T)}. \end{aligned}$$

(A.2b)

Here $\tau _T^* := \max \tau |_{\partial T \setminus F^*}$ and $F^*$ is a face of T at which $\tau |_{\partial T}$ is maximum. As is customary, the symbol $|\cdot |_{H^s}$ is to be understood as the Sobolev semi norm of order $s\in {\mathbb {R}}$.

Proof of Lemma 1

In this section we present the proof of Lemma 1, relating to the well posedness of the auxiliary non linear local problem that leads to the post processed approximation $u^*_h$. We re state the Lemma here for convenience.

Lemma 1

The local post processing $u_h^*$ is well defined for L small enough. Moreover, if $Lh^2<1$ and $k\ge 1$, then

$$\begin{aligned} \Vert u-u_h^*\Vert _{0,\mathcal {T}_h}&\lesssim (Rh)^{1/2} (h^{l_u+1} |u|_{l_u+2,\mathcal {T}_h} + h^{l_u+1} |\varvec{q}|_{l_{\varvec{q}}+2,\mathcal {T}_h} )+h^{l_u+2} |u|_{l_u+2,\mathcal {T}_h}\nonumber \\&\quad +L h^{l_u+1} |u|_{l_u+2,\mathcal {T}_h}, \end{aligned}$$

(B.1a)

$$\begin{aligned} |u-u_h^*|_{1,T}&\lesssim h_T^{l_u+1} |u|_{l_u+1,T} + Lh_T\Vert \varepsilon ^u \Vert _{0,T} +\Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T} + Lh_T\Vert u-u_h\Vert _{0,T}, \end{aligned}$$

(B.1b)

and

$$\begin{aligned} \sum _{e\in {\mathcal {E}}_h^\partial } h_e^{1/2} \Vert [ \! [ {u_h^*} ] \! ] \Vert _e&\lesssim \Vert u-u_h^*\Vert ^{1/2}_{0,\mathcal {T}_h} \left( \Vert u-u_h^*\Vert _{0,\mathcal {T}_h}^2 + h^2 |u-u_h^*|^2_{1,\mathcal {T}_h} \right) ^{1/4}. \end{aligned}$$

(B.1c)

Proof

We will prove first that the problem (2.3) is well posed. For this, we will use a fixed point argument. Let $T\in \mathcal {T}_h$. We define the operator $S: \mathbb {P}_{k+1}(T) \rightarrow \mathbb {P}_{k+1}(T)$ as $S (\zeta )= z$, where z is the only solution of

$$\begin{aligned} (\kappa \nabla z, \nabla w)_T&= -(q_h,\nabla w)_T + (\mathcal {F}(u_h),w)_T - (\mathcal {F}(\zeta ), w)_T,&\forall \ w \in \mathbb {P}_{k+1}(T), \end{aligned}$$

(B.2a)

$$\begin{aligned} (z,w)_T&= (u_h,w)_T ,&\forall \ w\in \mathbb {P}_0(T). \end{aligned}$$

(B.2b)

Note that S is surjective because (B.2) is well-posed. We will show now that S has a unique a fixed point and in that case it is the solution of (2.3). Let $\zeta _1,\zeta _2 \in \mathbb {P}_{k+1}(T)$ such that $S(\zeta _1)=z_1$ and $S(\zeta _2)=z_2$, with $z_1$ and $z_2$ satisfying (B.2). We observe that $\zeta _1-\zeta _2 \in \mathbb {P}_{k+1}(T)$ and

$$\begin{aligned} (\kappa \nabla (z_1-z_2), \nabla w)_T&= - (\mathcal {F}(\zeta _1)-\mathcal {F}(\zeta _2), w)_T,&\forall \ w \in \mathbb {P}_{k+1}(T), \end{aligned}$$

(B.3a)

$$\begin{aligned} (z_1-z_2,w)_T&= 0 ,&\forall \ w\in \mathbb {P}_0(T). \end{aligned}$$

(B.3b)

Then, for $i=1$ and 2, we set ${\overline{z}}_i:= \displaystyle \dfrac{1}{|T|} \int _T z_i$ and noticing that ${\overline{z}}_1={\overline{z}}_2$ by Eq. (B.3b), we have

$$\begin{aligned} \Vert z_1-z_2\Vert ^2_{T} = \Vert (z_1-\overline{z_1}) - (z_2-\overline{z_2}) \Vert ^2_{T} \le C_F^2 \Vert \kappa ^{1/2} \nabla (z_1-z_2)\Vert ^2_T, \end{aligned}$$

where we have used the Friedrichs inequality with constant $C_F>0$. Taking $w= z_1-z_2$ in (B.3a), and recalling that ${\mathcal {F}}$ is Lipschitz continuous with constant L, we obtain

$$\begin{aligned} \Vert z_1-z_2\Vert ^2_{T} \le C_F^2 ( \mathcal {F}(\zeta _2) - \mathcal {F}(\zeta _1) ,z_1-z_2)_T \le C_F^2L \Vert \zeta _2 - \zeta _1\Vert _{T} \Vert z_1-z_2\Vert _{T}. \end{aligned}$$

Thus, the operator S is a contraction as long as $C_F^2 L < 1$. If that is indeed the case, it has a unique fixed point.

For the inequality (B.1a), let $P_0$ and $P_{W^*}$ be the $L^2-$projectors into the space of constants and into $W_h^*$ respectively and decompose

$$\begin{aligned} u-u_h^* = (I-P_{W^*})u + P_0(P_{W^*}u - u_h^*) + (I-P_0)(P_{W^*}u-u_h^*) , \end{aligned}$$

(B.4)

We will now proceed to bound each of the terms on the right hand side of this expression separately in order to estimate the difference $u - u^*_h$. For the first term it is easy to see that

$$\begin{aligned} \Vert (I - P_{W^*})u\Vert _{0,T} \lesssim h_T^{l_u+2} |u|_{l_u+2,T}. \end{aligned}$$

(B.5)

For the second term we first notice that, since $W^*$ is a space of piecewise polynomials, the definitions of $P_{W^*}$ and $\Pi _{W}$, since $k\ge 1$, imply $P_0\,P_{W^*}u = P_0u = P_0\,\Pi _{W}u$

$$\begin{aligned} \Vert P_0(P_{W^* }u - u_h^*)\Vert _{0,T} = \Vert P_0(\Pi _{W} u - u_h)\Vert _{0,T} \le \Vert \Pi _{W} u - u_h\Vert _{0,T}= \Vert \varepsilon ^u \Vert _{0,T}.\nonumber \\ \end{aligned}$$

(B.6)

In the first equality we have made use of the fact that, due the definition of $u_h^*$ in Eq. (2.3b), we have $P_0 u_h^* = P_0 u_h$.

Now we move on to the third term in (B.4) and note that for every ${\varvec{v}}$ in the space of vector valued functions with components belonging to $W^*_h$ and $T\in {\mathcal {T}}$ it holds that

$$\begin{aligned} (\kappa \nabla (u-u_h^*), {\varvec{v}})_T =\, \left( \kappa \nabla \left( P_{W^*}u-u_h^*\right) , {\varvec{v}}\right) _T = (\kappa \nabla \,(I-P_0)(P_{W^*}u-u_h^*), {\varvec{v}})_T.\nonumber \\ \end{aligned}$$

(B.7)

Moreover, for the exact solutions $(u,{\varvec{q}})$, we have $\kappa \nabla u = -{\varvec{q}}$ so that the difference $u-u_h^*$ satisfies

$$\begin{aligned}&(\kappa \nabla (u-u_h^*), \nabla w)_T =\,- (\varvec{q} - \varvec{q}_h, \nabla w)_T\\&+ (\mathcal {F}(u_h^*) - \mathcal {F}(u),w )_T - (\mathcal {F}(u_h) - \mathcal {F}(u),w )_T \end{aligned}$$

for every $w\in W^*_h$ and $T\in {\mathcal {T}}$. Letting $w:=(I-P_0)(P_{W^*}u-u_h^*)\in W^*$ and $\nabla w$ be the test functions above, and using conditions (B.7) leads to

$$\begin{aligned} (\kappa \nabla w, \nabla w)_T = -(\varvec{q} - \varvec{q}_h, \nabla w)_T + (\mathcal {F}(u_h^*) - \mathcal {F}(u),w )_T + (\mathcal {F}(u)- \mathcal {F}(u_h),w)_T. \end{aligned}$$

From this equation, using the the scaling argument $\Vert w\Vert _{0,T} \lesssim h_T |w|_{1,T}$ and the inverse inequality $|w|_{1,T}\lesssim h_T^{-1} \Vert w\Vert _{0,T}$ we arrive at

$$\begin{aligned}&h_T^{-2}\, \Vert w\Vert ^{2}_{0,T}\lesssim {{\overline{\kappa }}}|w|^2_{1,T}\le \, \Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T}|w|_{1,T}\\&+ L\, \left( \Vert u-u^*_h\Vert _{0,T} + \Vert u-u_h\Vert _{0,T}\right) \Vert w\Vert _{0,T} \end{aligned}$$

from which we conclude that

$$\begin{aligned} \Vert w\Vert _{0,T}\lesssim h_T\, \Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T} + L \, h_T^2\, \left( \Vert u-u^*_h\Vert _{0,T} + \Vert u-u_h\Vert _{0,T}\right) . \end{aligned}$$

Recalling the decomposition (B.4), and the estimates (B.5), (B.6) we can bound the term $\Vert u-u^*_h\Vert _{0,T}$ on the right hand side yielding

$$\begin{aligned}&(1-Lh^2_T)\Vert w\Vert _{0,T} \lesssim h_T\, \Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T}\nonumber \\&+ L \, h_T^2\, \left( h_T^{l_u+2} |u|_{l_u+2,T} + \Vert \varepsilon ^u \Vert _{0,T} + \Vert u-u_h\Vert _{0,T}\right) \end{aligned}$$

(B.8)

Combining (B.8) above with (B.5) and (B.6) once more we arrive at

$$\begin{aligned} (1-\, Lh_T^2)\, \Vert u-u_h^*\Vert _{0,T}&\lesssim \, (1-Lh_T^2)h_T^{l_u+2} |u|_{l_u+2,T} + (1-Lh_T^2)\Vert \varepsilon ^u \Vert _{0,T} \\&\quad +h_T\, \Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T} \\&\quad +L \, h_T^2 \left( h_T^{l_u+2} |u|_{l_u+2,T} + \Vert \varepsilon ^u \Vert _{0,T} + \Vert u-u_h\Vert _{0,T}\right) \\&\lesssim \, h_T^{l_u+2} |u|_{l_u+2,T} + \Vert \varepsilon ^u \Vert _{0,T}\\&\quad +h_T\, \Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T} + L \, h_T^2\, \Vert u-u_h\Vert _{0,T}. \end{aligned}$$

So, assuming $Lh_T^2<1$ for each $T\in \mathcal {T}_h$, results

$$\begin{aligned} \Vert u-u_h^*\Vert _{0,T} \lesssim h_T^{l_u+2} |u|_{l_u+2,T} + \Vert \varepsilon ^u \Vert _{0,T} +h_T\, \Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T} + Lh_T^2\Vert u-u_h\Vert _{0,T}. \end{aligned}$$

By adding on each $T\in \mathcal {T}_h$, the estimate (B.1a) is concluded after considering the results in Theorem 2. Now, if we apply the inverse inequality to the estimate above, we arrive at

$$\begin{aligned} (1-Lh^2_T)|w|_{1,T} \lesssim \Vert {\varvec{q}} - {\varvec{q}}_h\Vert _{0,T} + L h_T\, \ \left( h_T^{l_u+2} |u|_{l_u+2,T} + \Vert \varepsilon ^u \Vert _{0,T} + \Vert u-u_h\Vert _{0,T}\right) . \end{aligned}$$

Assuming again $Lh_T^2<1$ for each $T\in \mathcal {T}_h$, (B.1b) follows.

Finally, using the trace inequality, the fact that $h_e \Vert v\Vert ^2_{0,e} \lesssim \Vert v\Vert _{0,T} \Big ( \Vert v\Vert ^2_{0,T} + h^2_T |v|^2_{1,T} \Big )^{1/2}$ for any $v \in [H^1(K)]^d$, and the estimates (B.1a) and (B.1b), we have

$$\begin{aligned} \sum _{e\in \mathcal {E}_h} h_e \Vert [ \! [ {u_h^*} ] \! ] \Vert ^2_{0,e}&\lesssim \sum _{e\in \mathcal {E}_h} \sum _{T'\in \omega _e} h_e \Vert u-u_h^*|_{T'}\Vert _{0,e}^2 \\&\lesssim \sum _{e\in \mathcal {E}_h} \sum _{T'\in \omega _e} \Vert u-u_h^*\Vert _{0,T'}\left( \Vert u-u_h^*\Vert _{0,T'}^2 + h_{T'}^2 |u-u_h^*|^2_{1,T'} \right) ^{1/2}, \end{aligned}$$

which implies (B.1c). $\square$

Auxiliary estimates

The following results were used throughout the text. We include them here for completeness.

The first lemma was needed to bound the terms in the decomposition of $\mathbb {T}^u$ carried out in Lemma 6.

Lemma C1

[9, Lemma 5.5] Suppose Assumption (3.2d) and the elliptic regularity inequality (3.10) hold. Then,

$$\begin{aligned} \Vert {(h^{\perp })}^{-1/2} (Id_M-P_M) \psi \Vert _{\Gamma _h}&\lesssim h \Vert \Theta \Vert _{\Omega }, \end{aligned}$$

(C.1a)

$$\begin{aligned} \Vert l^{1/2} (Id_M-P_M)\partial _n \psi \Vert _{\Gamma _h}&\lesssim R^{1/2} h \Vert \Theta \Vert _{\Omega }, \end{aligned}$$

(C.1b)

$$\begin{aligned} \Vert l^{-3/2} (\psi + l \partial _n \psi ) \Vert _{\Gamma _h}&\lesssim \Vert \Theta \Vert _{\Omega }, \end{aligned}$$

(C.1c)

$$\begin{aligned} \Vert l^{-1} \psi \Vert _{\Gamma _h}&\lesssim \Vert \Theta \Vert _{\Omega }. \end{aligned}$$

(C.1d)

The result below is used when deducing the bound for the term of the estimator involving the jump in the flux.

Lemma C2

Let $e\in \mathcal {E}^\partial _h$ and $\varvec{v}\in \varvec{H}(\mathbf{div} ;T^e)$. It holds

$$\begin{aligned} \Vert E_{T^e}(\varvec{v})\Vert ^2_{T^e_{ext}} \lesssim \,&r_e^2 \, \Vert \varvec{v}\Vert ^2_{T^e} + r_e^2\, h_T^2 \Vert \nabla \cdot \varvec{v}\Vert ^2_{T^e} . \end{aligned}$$

(C.2a)

Proof

We employ a scaling argument. Let $\Phi : T^e\rightarrow {\widehat{T}}$ be the affine mapping from $T^e$ to the reference element ${\widehat{T}}$ and set $\widehat{T_{ext}^e} := \Phi ^{-1}(T_{ext}^e)$. We have

$$\begin{aligned} \Vert E_{T^e}(v)\Vert ^2_{T^e_{ext}}&= 2|T^e_{ext}| \Vert \widehat{E}(\widehat{v})\Vert _{\widehat{T^e_{ext}}}^2 \lesssim |T^e_{ext}| \Vert \widehat{v}\Vert ^2_{H(\mathbf{div} ; {\widehat{T}})} = |T^e_{ext}| \left( \Vert \widehat{v} \Vert ^2_{\widehat{T}} + \Vert \widehat{\nabla } \cdot \widehat{v} \Vert ^2_{\widehat{T}} \right) \\&\lesssim |T^e_{ext}| \left( \dfrac{1}{|T^e|} \, \Vert v\Vert ^2_{T^e} + \Vert \nabla \cdot v\Vert ^2_{T^e} \right) . \end{aligned}$$

Thus, considering that $|T^e_{ext}| \lesssim (H_e^{\perp })^2 = R_e^2\, (h_e^{\perp })^2 \le r_e^2\, h_T^2$, and $|T^e|\lesssim h_T^2$, the inequality (C.2a) can be deduced. $\square$

The following result pertaining bubble functions is useful when addressing the local efficiency of the error estimator.

Lemma C3

[24, Lemma 3.3] Let $B_T:= \Pi _{i=1}^{d+1} \lambda _i$ be the element–bubble function associated to $T\in \mathcal {T}_h$, where $\{\lambda _i\}_{i=1}^{d+1}$ are the barycentric coordinates of T, and $B_e:= \Pi _{\begin{array}{c} i=1\\ i\ne j \end{array}}^{d+1} \lambda _i$ be the face–bubble function associated to $e\subset \partial T$, where $\lambda _j$ vanishes on e. Then, the following estimates hold

$$\begin{aligned} \begin{array}{rlcrlcrl} \Vert \varvec{v}\Vert ^2_T \lesssim &{} (\varvec{v}, B_T\varvec{v})_T, &{}\quad &{} \Vert B_T\varvec{v}\Vert _T \lesssim &{} \Vert \varvec{v}\Vert _T, &{}\quad &{} \Vert B_T\varvec{v}\Vert _{1,T} \lesssim &{} h_T^{-1}\, \Vert \varvec{v}\Vert _T, \\ \Vert \varvec{\mu }\Vert ^2_e \lesssim &{} (\varvec{\mu }, B_e\varvec{\mu })_e, &{}\quad &{} \Vert B_e\varvec{\mu }\Vert _{\Delta _e} \lesssim &{} h_e^{1/2}\, \Vert \varvec{\mu }\Vert _e, &{}\quad &{} \Vert B_e\varvec{\mu }\Vert _{1,\Delta _e}, \lesssim &{} h_e^{-1/2}\, \Vert \varvec{\mu }\Vert _e, \end{array} \end{aligned}$$

(C.3)

for all $\varvec{v}\in [\mathbb {P}_k(T)]^d, \ T\in \mathcal {T}_h$ and for each $\varvec{\mu }\in [\mathbb {P}_k(e)]^d, \ e\in \mathcal {E}_h$.

Clément and Oswald interpolants

The following two interpolants are useful in the arguments leading to the reliability of the estimator. They allow to control the behavior of functions with piecewise $H^1$ regularity by representatives belonging to the global $H^1_0(\Omega )$ space.

First, in the next lemma, we state the approximation properties of the Clément interpolation operator $\mathcal {C}_h: L^2(\Omega _h) \rightarrow W_h^{1,c} \cap H_0^1(\Omega )$, introduced in [5] as

$$\begin{aligned} \mathcal {C}_h w := \sum _{z\in \mathcal {N}_h} \left( \dfrac{1}{|\Omega _z|} \int _{\Omega _z} w \ dx \right) \phi _z \end{aligned}$$

where $\phi _z$ is the $\mathbb {P}_1$ nodal basic functions associated to the interior vertex z, $\Omega _z = supp \ \phi _z$, and $W_h^{1,c}:= \{ w\in C(\Omega ) : w|_T\in {\mathbb {P}}_1(T), T\in \mathcal {T}_h \}$.

Lemma D4

[24, Lemma 3.2] For any $T \in \mathcal {T}_h , \ e\in \mathcal {E}_h^i$ and $0\le m\le 1$, the following estimates hold, for all $w\in H_0^1(\Omega )$

$$\begin{aligned}&\Vert \mathcal {C}_h w \Vert _{m,\Omega } \lesssim \Vert w\Vert _{m,\Omega }, \quad \Vert w-\mathcal {C}_h w\Vert _{0,T} \lesssim h_T \Vert w\Vert _{1,\Delta _T},\\&\quad \Vert w-\mathcal {C}_h w\Vert _{0,e} \lesssim h_e^{1/2} \Vert w\Vert _{1,\Delta _e}, \end{aligned}$$

where $\Delta _T:= \{ T'\in \mathcal {T}_h : {\overline{T}} \cap \overline{T'} \ne \emptyset \}$ and $\Delta _e = \{T'\in \mathcal {T}_h: \overline{T'} \cap {\overline{e}} \ne \emptyset \}$.

The next results shows that an element w of $W_h^*$ can be approximated by a continuous function $\widetilde{w}\in W_h^*$, sometimes referred to as Oswald interpolant, and that the approximation error can be controlled by the size of the inter-element jumps of w.

Lemma D5

[20, Theorem 2.2] For any $w_h\in W_h^*$ and any multi-index with $|\alpha |=0,1$, the following approximation results holds: Let $u_D$ be the restriction to $\Gamma _h$ of a function in $W_h^*\cap H^1(\Omega _h)$. then there exists a function $\widetilde{w}_h\in W_h^*\cap H^1(\Omega _h)$ satisfying $\widetilde{w}_h|_{\Gamma } = u_D$, and

$$\begin{aligned} \sum _{T\in \mathcal {T}_h} \Vert D^{\alpha } (w_h - \widetilde{w}_h)\Vert ^2_T \le C_O \left( \sum _{e\in \mathcal {E}_h^\circ } h_e^{1-2|\alpha |} \Vert [ \! [ {w_h} ] \! ] \Vert ^2_e + \sum _{e\in \mathcal {E}_h^{\partial }} h_e^{1-2|\alpha |} \Vert u_D-w_h\Vert ^2_e\right) , \end{aligned}$$

above, $C_O$ is a positive constant independent of the mesh size.