Proof. For (2.3), we refer to [Reference Cummings and Feng17, Reference Hetmaniuk31]. On the other hand, the singularity decomposition comes form [Reference Chaumont-Frelet and Nicaise16, Reference Duan and Wu21].

Proof. We rewrite (2.2) as

\begin{align*} a_{0}(w_\psi ,v)=(\widetilde F,v)+\langle g,v\rangle _{\Gamma _{\textrm {imp}}}\text{ with }\widetilde F\,:\!=\,F+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}(2\left \vert \psi \right \vert ^2w_\psi +\psi ^2\overline {w_\psi }), \quad \forall v\in H^1_{\Gamma _{\textrm {Dir}}}(\Omega ). \end{align*}

Then by Lemma 2.1,

\begin{align*} \left\Vert {\hskip -4.1pt}\left\vert w_\psi \right\vert {\hskip -4.1pt}\right\Vert \lesssim M(\widetilde F,g)\lesssim M(F,g)+k\varepsilon \left \Vert \psi \right \Vert _{L^{\infty }(\Omega _{0})}^2k\left\Vert w_\psi \right\Vert _{0}\lesssim M(F,g)+\theta _0\left\Vert {\hskip -4.1pt}\left\vert w_\psi \right\vert {\hskip -4.1pt}\right\Vert , \end{align*}

which implies the first estimate in (2.4) if $\theta _0$ is sufficiently small. Moreover, $w_\psi$ has a decomposition $w_\psi =w_R+C^j_kS_j$ satisfying the estimates:

\begin{align*} \begin{aligned} \left \Vert w_R\right \Vert _{2}&\lesssim k\widehat {M}(\widetilde F,g)\lesssim k\widehat {M}(F,g)+kk\varepsilon \left \Vert \psi \right \Vert _{L^{\infty }(\Omega _{0})}^2k\left\Vert w_\psi \right\Vert _{0}\lesssim k\widehat {M}(F,g) \quad \text{and}\\ \big \vert C^j_k\big \vert &\lesssim k^{\alpha _j-\frac 12}M(\widetilde F,g)\lesssim k^{\alpha _j-\frac 12}M(F,g). \end{aligned} \end{align*}

That is, (2.5) holds.

Finally, the second estimate in (2.4) may be proved by following the proof of [Reference Wu and Zou47, Lemma 2.3] but an addition term $\big \|\frac {\partial w_\psi }{\partial n}\big \|_{0,\Gamma _{\textrm {Dir}}}$ appears, which can be estimated by following the proof of [Reference Hetmaniuk31, Proposition 3.3]. Here, we omit the details. The proof is completed.

Proof. We prove this lemma by induction. We let $C_t, C_{\infty }, C_r, C_s, C_t^0, C_\infty ^0$ be the hidden constants in (2.4), (2.5), and (2.7), respectively. Denote by $\widetilde C_t=\max \{2C_t,C_t^0\},\widetilde C_{\infty }=\max \{2C_{\infty },C_\infty ^0\},\widetilde C_r=2C_r$ and $\widetilde C_s=2C_s$ . Suppose $\varepsilon M(f,g)^2\leq \hat \theta \,:\!=\, \min \Big \{ \frac {\theta _0}{\widetilde C_{\infty }^2}, \frac 1{2\widetilde C_t \widetilde C_\infty ^2}\Big \}$ with $\theta _0$ from Lemma 2.2. Next, we prove by induction that (2.8) and (2.9) hold with hidden constants $\widetilde C_t, \widetilde C_{\infty }$ and $\widetilde C_r, \widetilde C_{s}$ , respectively. It suffices to prove them under the conditions:

(2.10) \begin{align} k\left \Vert u^{l-1}\right \Vert _{0,\Omega _0}\le \widetilde C_tM(f,g),\quad \left \Vert u^{l-1}\right \Vert _{L^{\infty }(\Omega _{0})}\le k^{-\frac 12}\widetilde C_{\infty }M(f,g), \end{align}

since (2.10) holds for $l=1$ . Using (2.10), we have

(2.11) \begin{align} M(f^{l-1},g)\leq (1+2\widetilde C_t\widetilde C_{\infty }^2\hat \theta )M(f,g)\leq 2M(f,g). \end{align}

Therefore, with the aid of Lemma 2.2, we can get

\begin{align*} &\left \Vert {\hskip -4.1pt}\left \vert u^l \right \vert {\hskip -4.1pt}\right \Vert \leq C_t M(f^{l-1},g)\leq \widetilde C_tM(f,g)\quad \mbox{and}\quad \left \Vert u^l\right \Vert _{L^{\infty }(\Omega _{0})}\leq k^{-\frac 12}C_{\infty }M(f^{l-1},g)\leq k^{-\frac 12}\widetilde C_{\infty }M(f,g), \end{align*}

and $u^l=u^l_{R}+\sum _{j=1}^N C_k^{j,l}S_j$ with the estimates:

\begin{align*} \left \Vert u^l_R\right \Vert _{2}\leq kC_r\widehat M(f^{l-1},g)\leq k\widetilde C_r\widehat {M}(f,g) \quad \mbox{and}\quad \big \vert C_k^{j,l}\big \vert \leq k^{\alpha _j-\frac 12}C_sM(f^{l-1},g)\leq k^{\alpha _j-\frac 12}\widetilde C_sM(f,g). \end{align*}

The proof is completed.

Proof. Let $v^l=u^{l+1}-u^l$ , through (2.6) and simple computations, we get for $l\geq 1$

\begin{align*} \begin{aligned} a_0(v^l,v)-k^2\varepsilon \big (2\left \vert u^l \right \vert v^l+(u^l)^2\overline {v^l,v}\big )_{\Omega _{0}}=k^2\varepsilon \big (((u^l)^2-(u^{l-1})^2)\overline {u^l-2}\left \vert u^{l-1} \right \vert ^2v^{l-1},v\big )_{\Omega _{0}} \quad \forall v\in H^1_{\Gamma _{\textrm {Dir}}}(\Omega ). \end{aligned} \end{align*}

Then with the aid of Lemma 2.2 and (2.8), we have

\begin{align*} \begin{aligned} \left \Vert {\hskip -4.1pt}\left \vert v^l \right \vert {\hskip -4.1pt}\right \Vert &\leq \widetilde C_t k^2\varepsilon \big \Vert u^l\big (|u^l|^2-|u^{l-1}|^2\big )\big \Vert _{0,\Omega _{0}}+2\widetilde C_tk^2\varepsilon \big \Vert |u^{l-1}|^2v^{l-1}\big \Vert _{0,\Omega _{0}}\\ &\leq \widetilde C_tk^2 \varepsilon \big \Vert u^l \big \Vert _{L^\infty (\Omega _{0})} \big \Vert |u^{l}|+|u^{l-1}| \big \Vert _{L^\infty (\Omega _{0})} \big \Vert v^{l-1} \big \Vert _{0} + 2\widetilde C_tk^2 \varepsilon \big \Vert u^{l-1} \big \Vert _{L^\infty (\Omega _{0})}^2 \big \Vert v^{l-1} \big \Vert _{0}\\ &\leq \widetilde C \varepsilon M(f,g)^2\left \Vert {\hskip -4.1pt}\left \vert v^{l-1} \right \vert {\hskip -4.1pt}\right \Vert \!. \end{aligned} \end{align*}

So we let $\varepsilon M(f,g)^2$ be sufficiently small to get

\begin{align*} \left \Vert {\hskip -4.1pt}\left \vert v^l \right \vert {\hskip -4.1pt}\right \Vert \leq \frac 12 \left \Vert {\hskip -4.1pt}\left \vert v^{l-1} \right \vert {\hskip -4.1pt}\right \Vert \!, \end{align*}

which means $\{u^l\}$ is a Cauchy sequence in the energy norm.

From the singularity decompositions in Lemmas 2.2–2.3, we know $v^l = u^{l+1}-u^{l} = u^{l+1}_R-u^l_R+\sum _{j=1}^N(C^{j,l+1}_k-C^{j,l}_k)S_j = v^l_R+\sum _{j=1}^N\widetilde C^{j,l}_kS_j$ and deduce

\begin{align*} \begin{aligned} \left \Vert v^l_R\right \Vert _{2} \lesssim k \left (k^2\varepsilon \big \Vert u^l\big (|u^l|^2-|u^{l-1}|^2\big )\big \Vert _{0,\Omega _{0}}+2 k^2\varepsilon \big \Vert |u^{l-1}|^2v^{l-1}\big \Vert _{0,\Omega _{0}}\right ) \leq C\theta _0 k k \left \Vert v^{l-1}\right \Vert _{0} \leq C\theta _0 k \left \Vert {\hskip -4.1pt}\left \vert v^{l-1} \right \vert {\hskip -4.1pt}\right \Vert \!, \end{aligned} \end{align*}

which implies the sequence $\{u^l_R\}$ is a Cauchy sequence in $H^2(\Omega )$ , then $u_R\,:\!=\,\lim \limits _{l\to \infty }u^l_R$ .

Similarly, we can get $\big \vert \widetilde C^{j,l}_k\big \vert \leq Ck^{\alpha _j-\frac 12}k\left \Vert v^{l-1}\right \Vert _{0}$ , then $\lim \limits _{l \to \infty } \widetilde C^{j,l}_k=0$ . Consequently, $C^j_k\,:\!=\,\lim \limits _{l \to \infty }C^{j,l}_k$ . Then, $u=u_R+\sum _{j=1}^NC^j_kS_j$ and the two components satisfying (2.13).

Finally considering uniqueness, we assume that there exists another solution of (2.1), denoted by $u^\ast$ . Since (2.6) holds for both $u$ and $u^\ast$ , letting $w=u-u^\ast$ we may obtain as above

\begin{equation*} \begin{aligned} a_0(w,v) - k^2 \varepsilon \big ( 2 \left \vert u \right \vert ^2 w + u^2 \overline {w,v}\big )_{\Omega _{0}} = k^2 \varepsilon \big (( u^2-(u^{\ast })^2 ) \overline {u^\ast } - 2 \left \vert u \right \vert ^2 w,v \big )_{\Omega _{0}} \quad \forall v\in H^1_{\Gamma _{\textrm {Dir}}}(\Omega ). \end{aligned} \end{equation*}

Then, the same argument as for $\left \Vert {\hskip -4.1pt}\left \vert v^l \right \vert {\hskip -4.1pt}\right \Vert \leq \frac 12 \left \Vert {\hskip -4.1pt}\left \vert v^{l-1} \right \vert {\hskip -4.1pt}\right \Vert$ implies that $\left\Vert {\hskip -4.1pt}\left\vert w \right\vert {\hskip -4.1pt}\right\Vert \leq \frac 12 \left\Vert {\hskip -4.1pt}\left\vert w \right\vert {\hskip -4.1pt}\right\Vert$ when $\varepsilon M(f,g)^2$ is sufficiently small. Thus, the uniqueness follows with $u=u^\ast$ .

Remark 2.5. (i) Theorem 2.4 states that under the condition $\varepsilon M(f,g)^2\leq \theta _0$ indicating that the nonlinearity may not be too strong, the NLH problem attains a unique low-energy solution satisfying the stability estimates and the singularity decomposition. However, this result does not exclude the possibility of multiple solutions to the NLH problem, nor does it cover the case of strong nonlinearity. The quadratic convergence of Newton’s iteration (2.6) when the initial guess is sufficiently close to the exact solution is proved in [ Reference Jiang, Li, Wu and Zou34 ], where the boundary condition is the truncated PML. For the impedance boundary condition, the proof is similar and the resulting quadratic convergence is valid for the high-energy solutions. The details are omitted.

(ii) Here are two other iteration methods. The first and simplest one is the frozen-nonlinearity iteration: given $u^0\in H^1_{\Gamma _{\mathrm {Dir}}}(\Omega )$ , find $u^{l+1}\in H^1_{\Gamma _{\mathrm {Dir}}}(\Omega )$ for $l=0,1,2,\ldots$ such that

\begin{align*} a_{u^{l}}(u^{l+1},v)=(f,v)+\langle g,v\rangle _{\Gamma _{\mathrm {imp}}}\quad \forall v\in H^1_{\Gamma _{\mathrm {Dir}}}(\Omega ). \end{align*}

In [ Reference Maier and Verfürth38, Reference Wu and Zou47 ], the linear convergence of the iteration sequence for the low-energy solution is derived, that is to say, this iteration converges when $\varepsilon M(f,g)^2$ is small enough.

Another one is the modified Newton’s method proposed in [ Reference Yuan and Lu48 ], which utilizes $\overline {u^{l+1}}$ in (2.6) instead of $\overline {u^l}$ : given $u^0\in H^1_{\Gamma _{\mathrm {Dir}}}(\Omega )$ , find $u^{l+1}\in H^1_{\Gamma _{\mathrm {Dir}}}(\Omega )$ for $l=0,1,2,\ldots$ such that

\begin{align*} a_{\sqrt 2u^{l}}(u^{l+1},v)=(f-k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u^l\vert ^2u^l,v)+\langle g,v\rangle _{\Gamma _{\mathrm {imp}}}\quad \forall v\in H^1_{\Gamma _{\mathrm {Dir}}}(\Omega ). \end{align*}

Numerical results show that the modified Newton’s method is only linearly convergent but robust with respect to the initial guess.

Proof. In this proof, we introduce a new elliptic projection $P_h^\gamma \colon V\mapsto V_{\mathcal{T}}$ (see, e.g., [Reference Du and Wu20, (6.9)]), defined by

\begin{align*} b^\gamma (P_h^\gamma \phi , v_h)=b^\gamma (\phi ,v_h),\quad \forall v_h\in V_{\mathcal{T}},\; \phi \in V. \end{align*}

Similar to (2.19), the following estimate holds:

(2.25) \begin{align} h_{\mathcal{T}}^{-\alpha }\left \Vert \phi -P_h^\gamma \phi \right \Vert _{0}+h_{\mathcal{T}}^{-\frac {\alpha }2}\left \Vert \phi -P_h^\gamma \phi \right \Vert _{0,\Gamma _{\textrm {imp}}} \lesssim \|\phi -P_h^\gamma \phi \|_J \lesssim \inf _{v_h\in V_{\mathcal{T}}}\|\phi -v_h\|_J, \end{align}

where $\|\cdot \|_J\,:\!=\,\big (\|\cdot \|_1^2 + |\cdot |_J^2 \big )^{\frac 12}$ .

The proof of the first estimate in (2.24) is similar to that of [Reference Wu and Zou47, Lemma 3.2] with the help of (2.25) and (2.17) due to corner singularities. We prove it in the appendix.

Next we prove the second estimate. First, by using the singularity decomposition of $w_\psi$ in Lemma 2.2 and following the proof of [Reference Jiang, Li, Wu and Zou34, Lemma 4.1], which uses the modified duality argument and regards the second term in (2.15) as a small perturbation, we can derive the following error estimates in energy and $L^2$ norms by the new elliptic projection (see also [Reference Chaumont-Frelet and Nicaise16, Theorem 5.5], [Reference Wu46, Theorem 4.4]), under the condition that $\Re \gamma \geq -\gamma _1, |\gamma |\lesssim 1$ , $k\varepsilon \left \Vert \psi \right \Vert _{L^{\infty }(\Omega _{0})}^2\le \theta _0$ and $k^3h_{\mathcal{T}}^{1+\alpha }\le C_0$ , the details are omitted.

(2.26) \begin{align} \begin{split} \left\Vert {\hskip -4.1pt}\left\vert w_\psi -w_h \right\vert {\hskip -4.1pt}\right\Vert &\lesssim \big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^{\alpha }+k^3h_{\mathcal{T}}^2\big )\widehat {M}(F,g), \\ \left \Vert w_\psi -w_h\right \Vert _{0}&\lesssim \big (k^{-1}(kh_{\mathcal{T}})^{2\alpha }+k^2h_{\mathcal{T}}^2\big )\widehat {M}(F,g). \end{split} \end{align}

On the other hand, from [Reference Bramble and Xu10, Lemma 2.3], there holds

(2.27) \begin{align} \left \Vert w_h\right \Vert _{L^{\infty }(\Omega _{0})}\lesssim \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12}\left \Vert w_h\right \Vert _{1,\Omega _{0}}\!. \end{align}

Indeed, the result there can be easily verified for the non-quasi-uniform meshes by replacing $\left \vert \ln h \right \vert ^{\frac 12}$ with $\left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12}$ .

To proceed, let $\zeta _h=w_h-P_h^\gamma w_\psi$ , then we have

(2.28) \begin{align} \left \Vert w_h\right \Vert _{L^{\infty }(\Omega _{0})}\leq \left\Vert w_\psi \right\Vert _{L^{\infty }(\Omega _{0})}+\left \Vert w_\psi -I_h w_\psi \right \Vert _{L^{\infty }(\Omega _{0})}+\left \Vert I_h w_\psi -P_h^\gamma w_\psi \right \Vert _{L^{\infty }(\Omega _{0})}+\left \Vert \zeta _h\right \Vert _{L^{\infty }(\Omega _{0})}\!, \end{align}

where $I_h$ is the standard FE interpolation operator. The first term can be estimated by directly using (2.4). Noting that $w_\psi |_{\Omega _{0}}=w_R|_{\Omega _{0}}$ since $w_\psi =w_{R}+\sum _{j=1}^N C_k^jS_j$ and the support of $S_j(1\leq j\leq N)$ does not intersect $\Omega _{0}$ , so the second term is estimated by using Lemma 2.2 and the interpolation error estimate $\left \Vert w_R-I_hw_R\right \Vert _{L^\infty (\Omega _{0})}\lesssim h_{\mathcal{T}}\left \vert w_R\right \vert _{2,\Omega _{0}}$ (see, e.g., [Reference Brenner and Scott11, (4.4.8)]):

(2.29) \begin{align} \begin{aligned} \left \Vert w_\psi -I_h w_\psi \right \Vert _{L^{\infty }(\Omega _{0})}&=\left \Vert w_R-I_h w_R\right \Vert _{L^{\infty }(\Omega _{0})}\lesssim kh_{\mathcal{T}}\widehat {M}(F,g)= k^{-\frac 12}(k^3h_{\mathcal{T}}^2)^{\frac 12}\widehat {M}(F,g). \end{aligned} \end{align}

To estimate the third term, we apply (2.27), (2.25), (2.17), the interpolation estimates, the local trace inequality and Lemma 2.2 to obtain

(2.30) \begin{align} \begin{aligned} \left \Vert I_h w_\psi -P_h^\gamma w_\psi \right \Vert _{L^{\infty }(\Omega _{0})} &\lesssim \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12}\left \Vert I_h w_\psi -P_h^\gamma w_\psi \right \Vert _{1,\Omega _{0}}\\ &\lesssim \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12}\big (\!\left \Vert w_R-I_h w_R\right \Vert _{1,\Omega _{0}}+\left \Vert w_\psi -P_h^\gamma w_\psi \right \Vert _{1}\!\big )\\ &\lesssim \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12} \big ( \!\left \Vert w_R-I_h w_R\right \Vert _{1,\Omega _{0}} + \|w_\psi -\mathcal I_h w_\psi \|_J \big ) \\ &\lesssim \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12} k^{-\frac 12}\big ((k^3h_{\mathcal{T}}^2)^{\frac 12}+(kh_{\mathcal{T}})^\alpha \big )\widehat {M}(F,g). \end{aligned} \end{align}

It remains to estimate $\left \Vert \zeta _h\right \Vert _{L^{\infty }(\Omega _{0})}$ . By simple calculations, we find

\begin{align*} b^\gamma (\zeta _h,v_h) & =\big ((k^2+1+2k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert \psi \vert ^2)(w_h-w_\psi )+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\psi ^2(\overline {w_h}-\overline {w_\psi }),v_h\big )\\& \quad +\mathbf{i} k\langle w_\psi -w_h,v_h\rangle _{\Gamma _{\textrm {imp}}} \ \ \forall v_h\in V_{\mathcal{T}}. \end{align*}

Then we define $\zeta \in H^1_{\Gamma _{\textrm {Dir}}}(\Omega )$ as the solution of the following variational problem:

(2.31) \begin{align} b(\zeta ,v) & =\big ((k^2+1+2k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert \psi \vert ^2)(w_h-w_\psi )+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\psi ^2(\overline {w_h}-\overline {w_\psi }),v\big )\nonumber\\ & \quad +\mathbf{i} k\langle w_\psi -w_h,v\rangle _{\Gamma _{\textrm {imp}}} \ \ \forall v\in H^1_{\Gamma _{\textrm {Dir}}}(\Omega ). \end{align}

It is clear that $\zeta _h=P_h^{\gamma } \zeta$ . By the singularity decomposition of the elliptic equations in [Reference Grisvard27], we derive $\zeta =\zeta _R+\sum _{j=1}^Nc_jS_j$ , and

(2.32) \begin{align} \begin{aligned} \left \Vert \zeta _R\right \Vert _{2}+\sum _{j=1}^N\left \vert c_j \right \vert &\lesssim \big \Vert (k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert \psi \vert ^2)(w_\psi -w_h)\big \Vert _0+k\left \Vert w_\psi -w_h\right \Vert _{\frac 12,\Gamma _{\textrm {imp}}}\\ &\lesssim \big (k^2(k^{-1}(kh_{\mathcal{T}})^{2\alpha }+k^2h^2_{\mathcal{T}})+k(kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^{\alpha }+k^3h^2_{\mathcal{T}})\big )\widehat {M}(F,g)\\ &\lesssim (k^4h^2_{\mathcal{T}}+k^2h_{\mathcal{T}}+k^{\frac 12}(kh_{\mathcal{T}})^{\alpha })\widehat {M}(F,g), \end{aligned} \end{align}

where we use (2.26) and the trace inequality. Then, taking (2.32) into (2.25) and using (2.17), we have

(2.33) \begin{align} \left \Vert \zeta -\zeta _h\right \Vert _{0}+h_{\mathcal{T}}^{\alpha }\!\left \Vert \zeta -\zeta _h\right \Vert _{1}\lesssim h^{2\alpha }_{\mathcal{T}}(k^4h^2_{\mathcal{T}}+k^2h_{\mathcal{T}}+k^{\frac 12}(kh_{\mathcal{T}})^{\alpha })\widehat {M}(F,g). \end{align}

Like the analyses in (2.29) and (2.30), we get

(2.34) \begin{align} \begin{aligned} \left \Vert \zeta -\zeta _h\right \Vert _{L^{\infty }(\Omega _{0})}&\lesssim \left \Vert \zeta -I_h \zeta \right \Vert _{L^{\infty }(\Omega _{0})}+\left \Vert \zeta _h-I_h\zeta \right \Vert _{L^{\infty }(\Omega _{0})}\\ &\lesssim h_{\mathcal{T}}\left \vert \zeta _R\right \vert _{2}+\left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12}\big (\|\zeta -I_h \zeta \|_{1,\Omega _{0}} + \left \Vert \zeta -\zeta _h\right \Vert _{1}\!\big )\\ &\lesssim h_{\mathcal{T}}\left \vert \zeta _R\right \vert _{2}+\left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12} \big ( h_{\mathcal{T}}\left \vert \zeta _R\right \vert _{2} + \left \Vert \zeta - \zeta _h\right \Vert _{1} \!\big )\\ &\lesssim h^{\alpha }_{\mathcal{T}}(k^4h^2_{\mathcal{T}}+k^2h_{\mathcal{T}}+k^{\frac 12}(kh_{\mathcal{T}})^{\alpha })(1+\left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12}\!)\widehat {M}(F,g)\\ &\le k^{-\frac 12}\big (k^{\frac {3}{2}} h_{\mathcal{T}}+2k^{-\frac 12}\big )t(k,h_{\mathcal{T}})(1+\left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12}\!)\widehat {M}(F,g). \end{aligned} \end{align}

After rewriting (2.31) as:

\begin{align*} a_{0}(\zeta ,v)-k^2\varepsilon \big (2\left \vert \psi \right \vert ^2\zeta +\psi ^2\bar \zeta ,v\big )_{\Omega _{0}}&= \big ((k^2+1+2k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert \psi \vert ^2)(P_h^\gamma w_\psi - w_{\psi } + \zeta _h - \zeta ),v\big )\\ &\quad +k^2\varepsilon \big (\vert \psi \vert ^2\overline {(P_h^\gamma w_\psi -w_{\psi }+\zeta _h-\zeta )},v\big )_{\Omega _{0}}\\ &\quad +\mathbf{i} k\langle w_{\psi } - P_h^\gamma w_{\psi }+\zeta -\zeta _h,v\rangle _{\Gamma _{\textrm {imp}}} \quad \forall v\in H^1_{\Gamma _{\textrm {Dir}}}(\Omega ), \end{align*}

it follows from Lemma 2.2, (2.25), (2.33), the trace inequality, and the condition $k^3h_{\mathcal{T}}^{1+\alpha }\leq C_0$ that

(2.35) \begin{align} \begin{aligned} \left \Vert \zeta \right \Vert _{L^{\infty }(\Omega _{0})}&\lesssim k^{-\frac 12}\big (k^2\!\left \Vert P_h^\gamma w_\psi - w_{\psi } + \zeta _h - \zeta \right \Vert _{0}+k\!\left \Vert P_h^\gamma w_\psi - w_{\psi } + \zeta _h - \zeta \right \Vert _{0,\Gamma _{\textrm {imp}}}\!\big )\\ &\lesssim k^{-\frac 12}\big (k^2h_{\mathcal{T}}^\alpha (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha )+k^2h^{2\alpha }_{\mathcal{T}}(k^4h^2_{\mathcal{T}}+k^2h_{\mathcal{T}}+k^{\frac 12}(kh_{\mathcal{T}})^{\alpha })\\ &\quad +kh_{\mathcal{T}}^{\frac {\alpha }2}(kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha )+kh^{\frac {3\alpha }2}_{\mathcal{T}}(k^4h^2_{\mathcal{T}}+k^2h_{\mathcal{T}}+k^{\frac 12}(kh_{\mathcal{T}})^{\alpha })\big )\widehat {M}(F,g)\\ & \lesssim k^{-\frac 12}\big (t(k,h_{\mathcal{T}})+t^2(k,h_{\mathcal{T}})\big )\widehat {M}(F,g). \end{aligned} \end{align}

Finally, the result comes from combining (2.28), (2.4), (2.29), (2.34), (2.35), (2.23), and the fact $\frac 12\lt \alpha \le 1$ .

Proof. (2.36) can be directly obtained from the convergence of iteration methods, which can be proved by following the proof of Theorem2.4. We omit the details for simplicity. The proof of (2.37) is a little more complex, but it is still based on the convergences of $\{u^l\}$ and $\{u^l_h\}$ .

We define $\hat u_h^0=u_h^0$ and $\hat u_h^l\in V_{\mathcal{T}} (l=1,2,\ldots )$ to be the solution to the following problem

\begin{align*} a_0^\gamma (\hat u^l_h, v_h)-k^2\varepsilon \big (2\left \vert u^{l-1} \right \vert ^2\hat u^l_h+(u^{l-1})^2\overline {\hat u^l_h},v_h\big )_{\Omega _{0}}=(f^{l-1}, v_h)+\langle g, v_h\rangle _{\Gamma _{\textrm {imp}}}\quad \forall v_h\in V_{\mathcal{T}}. \end{align*}

Then, from (2.26) and (2.11), we have

(2.38) \begin{align} \begin{aligned} \left\Vert {\hskip -4.1pt}\left\vert u^l-\hat u^l_h \right\vert {\hskip -4.1pt}\right\Vert &\lesssim (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^{\alpha }+k^3h_{\mathcal{T}}^2)\widehat {M}(f,g),\\ k\left \Vert u^l-\hat u^l_h\right \Vert _{0}&\lesssim \big ((kh_{\mathcal{T}})^{2\alpha }+k^3h_{\mathcal{T}}^2\big )\widehat {M}(f,g). \end{aligned} \end{align}

Next we need to estimate $\hat \eta ^l_h\,:\!=\,\hat u^l_h-u^l_h$ . Through simple calculation, we find $\hat \eta _h^l$ satisfies

\begin{align*} a_0^\gamma (\hat \eta _h^l,v_h)-k^2\varepsilon \big (2\left \vert u^{l-1} \right \vert ^2\hat \eta _h^l+(u^{l-1})^2\overline {\hat \eta _h^l},v_h\big )_{\Omega _{0}}&=k^2\varepsilon \big (2\big (\!\left \vert u^{l-1} \right \vert ^2-\left \vert u^{l-1}_h \right \vert ^2\big )u^l_h+\big ((u^{l-1})^2-(u^{l-1}_h)^2\big )\overline {u^l_h},v_h\big )_{\Omega _{0}}\\ &\quad + 2k^2\varepsilon \big (\!\left \vert u^{l-1}_h \right \vert ^2u^{l-1}_h-\left \vert u^{l-1} \right \vert ^2u^{l-1},v_h\big )_{\Omega _{0}}\qquad \forall v_h\in V_{\mathcal{T}}. \end{align*}

It is shown from Lemmas 2.6, 2.3, and 2.7 that

\begin{align*} \left\Vert {\hskip -4.1pt}\left\vert \hat \eta ^l_h \right\vert {\hskip -4.1pt}\right\Vert &\lesssim k^2\varepsilon \left \Vert \big (\!\left \vert u^{l-1} \right \vert ^2-\left \vert u^{l-1}_h \right \vert ^2\!\big )(u^l_h-u^{l-1}_h)+\big ((u^{l-1})^2-(u^{l-1}_h)^2\big )\overline {u^l_h}+\left \vert u^{l-1} \right \vert ^2(u^{l-1}_h-u^{l-1})\right \Vert _{0,\Omega _{0}}\\ &\lesssim k\varepsilon \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert \widehat {M}(f,g)^2\left \Vert u^{l-1}-u^{l-1}_h\right \Vert _{0}. \end{align*}

So when $\varepsilon \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert \widehat {M}(f,g)^2$ is sufficiently small, we have

\begin{align*} \left\Vert {\hskip -4.1pt}\left\vert \hat \eta ^l_h \right\vert {\hskip -4.1pt}\right\Vert \leq \frac 12\left\Vert {\hskip -4.1pt}\left\vert u^{l-1}-\hat u^{l-1}_h \right\vert {\hskip -4.1pt}\right\Vert +\frac 12\left\Vert {\hskip -4.1pt}\left\vert \hat \eta ^{l-1}_h \right\vert {\hskip -4.1pt}\right\Vert \quad \mbox{and}\quad \left \Vert \hat \eta ^l_h\right \Vert _{0}\leq \frac 12\left \Vert u^{l-1}-\hat u^{l-1}_h\right \Vert _{0}+\frac 12\left \Vert \hat \eta ^{l-1}_h\right \Vert _{0}\!, \end{align*}

and then

(2.39) \begin{align} \left\Vert {\hskip -4.1pt}\left\vert \hat \eta ^l_h \right\vert {\hskip -4.1pt}\right\Vert \leq \sum _{j=0}^{l-1}2^{j-l}\left\Vert {\hskip -4.1pt}\left\vert u^j-\hat u^j_h \right\vert {\hskip -4.1pt}\right\Vert \lesssim (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^{\alpha }+k^3h_{\mathcal{T}}^2)\widehat {M}(f,g)+2^{-l}\left\Vert {\hskip -4.1pt}\left\vert u^0-u^0_h \right\vert {\hskip -4.1pt}\right\Vert . \end{align}

Combining (2.38) and (2.39), we derive

\begin{align*} \left\Vert {\hskip -4.1pt}\left\vert u^l-u^l_h \right\vert {\hskip -4.1pt}\right\Vert \lesssim (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^{\alpha }+k^3h_{\mathcal{T}}^2)\widehat {M}(f,g)+2^{-l}\left\Vert {\hskip -4.1pt}\left\vert u^0-u^0_h \right\vert {\hskip -4.1pt}\right\Vert . \end{align*}

Similarly,

\begin{align*} k\left \Vert u^l-u^l_h\right \Vert _{0}\lesssim \big ((kh_{\mathcal{T}})^{2\alpha }+k^3h_{\mathcal{T}}^2\big )\widehat {M}(f,g)+2^{-l}\left \Vert u^0-u^0_h\right \Vert _{0}\!. \end{align*}

Then, (2.37) comes from letting $l\rightarrow \infty$ in the above two estimates. This completes the proof of the theorem.

Remark 2.9. (i) Denote by $\mathcal{T}_0$ the initial mesh. $\vert \ln h_{\mathcal{T},\mathrm {min}} \vert / \vert \ln h_{\mathcal{T}_0} \vert \eqsim$ the maximum number of times the mesh is refined locally, that is to say, although the condition here is more stringent than the one in [ Reference Jiang, Li, Wu and Zou34 ] ( $\varepsilon \widehat {M}(f,g)^2\leq \theta _0$ for the impedance boundary condition), it is also acceptable. In addition, to the best of our knowledge, little is known about an $L^{\infty }$ interior estimate of $u_h$ for the NLH problem on a shape regular mesh, especially a locally refined mesh. Moreover, another technique for analyzing the case of a shape regular mesh is directly using the $H^{1}$ global estimate instead of the $L^{\infty }$ interior estimate and requires more stringent conditions (see, e.g. [ Reference Maier and Verfürth38 ]).

(ii) Similar to Remark 2.5 (i), Theorem 2.8 states that if $\left \vert \ln h_{\mathcal{T},\mathrm {min}} \right \vert \varepsilon \widehat {M}(f,g)^2\leq \theta _0$ and $k^3h_{\mathcal{T}}^{1+\alpha }\leq C_0$ , CIPFEM (2.14) attains a unique low-energy solution $u_h$ and $u_h$ approximates the low-energy solution $u$ to the NLH problem (2.1). Like the proof in [ Reference Jiang, Li, Wu and Zou34 ], we can prove the local quadratic convergence of Newton’s iteration (2.16), which also holds for the high-energy solutions. The details are omitted.

Remark 3.1. (i) Our error estimator $\eta _T$ on an element $T$ is slightly different from the standard residual-type error estimator defined as $\eta _T^{\mathrm {std}}\,:\!=\,\big (h_T^2\!\left \Vert R_{T}\right \Vert _{0,T}^2+h_T\left \Vert R_{\partial T}\right \Vert _{0,\partial T}^2\!\big )^{\frac {1}{2}}$ , but, apparently, their corresponding error estimators on the whole mesh are equivalent, that is, $\eta _{\mathcal{T}}\eqsim \eta _{\mathcal{T}}^{\mathrm {std}}$ . We introduce this new error estimator for the purpose of proving the convergence and quasi-optimality of the standard adaptive algorithm.

(ii) The auxiliary error estimators $\widetilde {\eta }_T$ associated with the elliptic projection are introduced only for the theoretical analysis and will not be used in the adaptive algorithm.

Proof. It follows from the local trace inequality and (2.17) that

\begin{align*} \left \vert J(u_h,\mathcal I_h\phi ) \right \vert &\lesssim |\gamma |\sum _{e\in \mathcal E^I}h_e\big \|[\![\nabla u_h]\!]\big \|_{0,e}\big \|[\![\nabla \mathcal I_h\phi ]\!]\big \|_{0,e}\\ &\lesssim |\gamma |\eta _{\mathcal{T}}\|\nabla \mathcal I_h\phi \|_0 \lesssim |\gamma |\eta _{\mathcal{T}}\|\nabla \phi \|_0. \end{align*}

The proof of the estimate for $|J(u_h,P_h \phi )|$ is analogous and is therefore omitted.

Proof. Denote by $e\,:\!=\,u-u_h$ , $\rho \,:\!=\,u-P_hu$ , and $\zeta _h\,:\!=\,P_hu-u_h$ . We show the following identity, which we will often use subsequently: for any $\phi \in H^1_{\Gamma _{\textrm {Dir}}}(\Omega )$ and $v_h\in V_{\mathcal{T}}$ ,

(3.8) \begin{align} a_{u}(e,\phi )=\sum _{T\in \mathcal{T}}\big ((R_T,\phi -v_h)_T+\langle R_{\partial T},\phi -v_h\rangle _{\partial T}\big ) +J(u_h,v_h)+k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,\phi \big )_{\Omega _{0}}. \end{align}

Indeed, with the aid of elementwise integration by parts and $a_u(u,v_h)-a_{u_h}^\gamma (u_h,v_h)=0$ by (2.1), (2.14), we derive

\begin{align*} a_{u}(e,\phi )&=a_{u}(u,\phi -v_h)-a_{u_h}(u_h,\phi -v_h) +J(u_h,v_h) +\big (a_{u_h}(u_h,\phi )-a_u(u_h,\phi )\big )\\ &=\big (f+k^2(1+\varepsilon {\mathbf{1}}_{\Omega _{0}}\left \vert u_h \right \vert ^2\!)u_h,\phi -v_h\big )+\langle g,\phi -v_h\rangle _{\Gamma _{\textrm {imp}}}-\mathbf{i} k\langle u_h,\phi -v_h\rangle _{\Gamma _{\textrm {imp}}}\\ & \quad \; -\sum _{T\in \mathcal{T}}\left\langle \frac {\partial u_h}{\partial n},\phi -v_h\right\rangle _{\partial T} +J(u_h,v_h) +k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,\phi \big )_{\Omega _{0}}\\ &=\sum _{T\in \mathcal{T}}\big ((R_T,\phi -v_h)_T+\langle R_{\partial T},\phi -v_h\rangle _{\partial T}\big )+J(u_h,v_h) +k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,\phi \big )_{\Omega _{0}}. \end{align*}

First, we estimate $\zeta _h$ . Taking $\phi =v_h=\zeta _h$ in (3.8), and using (2.20) and (2.18), we have

(3.9) \begin{align} a_u(\zeta _h,\zeta _h)&=a_u(e,\zeta _h)-a_u(\rho ,\zeta _h)\notag \\ &=k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h+\left \vert u \right \vert ^2\rho ,\zeta _h\big )_{\Omega _{0}}+J(u_h,\zeta _h) +(k^2+1)(\rho ,\zeta _h)-\mathbf{i} k\langle \rho ,\zeta _h\rangle _{\Gamma _{\textrm {imp}}}. \end{align}

Taking the imaginary part of (3.9) and noting $e = \rho + \zeta _h$ , by $L^\infty$ interior estimates in (2.12) and (2.36), we have

\begin{align*} \begin{aligned} k\left \Vert \zeta _h\right \Vert ^2_{0,\Gamma _{\textrm {imp}}}&\lesssim k\theta _0(\!\left \Vert e\right \Vert _{0}+\Vert \rho \Vert _{0})\left \Vert \zeta _h\right \Vert _{0}+k^2h_{\mathcal{T}}^\alpha \Vert \rho \Vert _{1}\left \Vert \zeta _h\right \Vert _{0}+k^{\frac 12}h_{\mathcal{T}}^{\frac {\alpha }2}\Vert \rho \Vert _{1}k^{\frac 12}\left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}} +|\gamma |\eta _{\mathcal{T}} \|\nabla \zeta _h\|_0 \\ &\lesssim k \left \Vert \zeta _h\right \Vert _{0}^2 +k^2h_{\mathcal{T}}^\alpha \Vert \rho \Vert _{1}\left \Vert \zeta _h\right \Vert _{0}+k^{\frac 12}h_{\mathcal{T}}^{\frac {\alpha }2}\Vert \rho \Vert _{1}k^{\frac 12}\left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}} +|\gamma |\eta _{\mathcal{T}} \|\nabla \zeta _h\|_0 \end{aligned} \end{align*}

where (2.19) and the local trace inequality are also used. Then, by Young’s inequality, we proceed

\begin{align*} k\left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}}^2\lesssim (k+k^3h_{\mathcal{T}}^{\alpha })\left \Vert \zeta _h\right \Vert _{0}^2+kh_{\mathcal{T}}^\alpha \Vert \rho \Vert _{1}^2 +|\gamma |\eta _{\mathcal{T}} \|\nabla \zeta _h\|_0, \end{align*}

which means

(3.10) \begin{align} \left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}}\lesssim (1+kh_{\mathcal{T}}^{\frac {\alpha }2})\left \Vert \zeta _h\right \Vert _{0}+h_{\mathcal{T}}^{\frac {\alpha }2}\Vert \rho \Vert _{1} +k^{-\frac 12}|\gamma |^{\frac 12}\eta _{\mathcal{T}}^{\frac 12} \|\nabla \zeta _h\|_0^{\frac 12}. \end{align}

Taking the real part of (3.9), we have the following inequalities by using (2.19) and (3.10)

\begin{align*} \left \Vert \nabla \zeta _h\right \Vert _{0}^2&\lesssim (k^2+\theta _0 k)\left \Vert \zeta _h\right \Vert _{0}^2+k\theta _0 h_{\mathcal{T}}^\alpha \Vert \rho \Vert _{1}\left \Vert \zeta _h\right \Vert _{0}+k^2h_{\mathcal{T}}^\alpha \Vert \rho \Vert _{1}\left \Vert \zeta _h\right \Vert _{0}+kh^{\frac {\alpha }2}_{\mathcal{T}}\Vert \rho \Vert _{1}\left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}} +|\gamma |\eta _{\mathcal{T}} \|\nabla \zeta _h\|_0 \\ &\lesssim k^2\left \Vert \zeta _h\right \Vert _{0}^2+(kh^{\frac {\alpha }2}_{\mathcal{T}}+k^2h_{\mathcal{T}}^\alpha )\Vert \rho \Vert _{1}\left \Vert \zeta _h\right \Vert _{0}+kh_{\mathcal{T}}^\alpha \Vert \rho \Vert _{1}^2 +|\gamma |^{\frac 12}k^{\frac 12}h^{\frac {\alpha }2}_{\mathcal{T}}\|\rho \|_1\eta _{\mathcal{T}}^{\frac 12} \|\nabla \zeta _h\|_0^{\frac 12} +|\gamma |\eta _{\mathcal{T}} \|\nabla \zeta _h\|_0 \\ &\lesssim k^2(1+kh_{\mathcal{T}}^\alpha )\left \Vert \zeta _h\right \Vert _{0}^2+kh_{\mathcal{T}}^\alpha \Vert \rho \Vert _{1}^2 +|\gamma |\eta _{\mathcal{T}} \|\nabla \zeta _h\|_0. \end{align*}

Consequently, by $kh_{\mathcal{T}}^\alpha \lesssim 1$ (see (2.23)), we derive

(3.11) \begin{align} \left \Vert \nabla \zeta _h\right \Vert _{0}\lesssim k\left \Vert \zeta _h\right \Vert _{0}+k^{\frac 12}h_{\mathcal{T}}^{\frac {\alpha }2}\Vert \rho \Vert _{1} +|\gamma |\eta _{\mathcal{T}}, \end{align}

which together with (3.10) implies that

(3.12) \begin{align} \left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}}\lesssim (1+kh_{\mathcal{T}}^{\frac {\alpha }2})\left \Vert \zeta _h\right \Vert _{0}+h_{\mathcal{T}}^{\frac {\alpha }2}\Vert \rho \Vert _{1} +k^{-\frac 12}|\gamma |\eta _{\mathcal{T}}+|\gamma |^{\frac 12}\eta _{\mathcal{T}}^{\frac 12} \|\zeta _h\|_0^{\frac 12}. \end{align}

Next, for the sake of estimating $\left \Vert \zeta _h\right \Vert _{0}$ , we consider the duality problem: find $z\in H^1_{\Gamma _{\textrm {Dir}}}(\Omega )$ such that

(3.13) \begin{align} a_{u}(v,z)=(v,\zeta _h)\quad \forall v\in H^1_{\Gamma _{\textrm {Dir}}}(\Omega ). \end{align}

By the similar proof in Lemma 2.2, we can derive the stability estimates and singularity decomposition for the above problem. Then with the help of (2.17), we have

(3.14) \begin{equation} k\left \Vert z\right \Vert _{0}\lesssim \left \Vert \zeta _h\right \Vert _{0},\quad \left \Vert z-\mathcal I_hz\right \Vert _{1} +|\mathcal I_hz|_J\lesssim (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha )\left \Vert \zeta _h\right \Vert _{0}. \end{equation}

Taking $v=\zeta _h$ in (3.13) and arguing as in (3.9), we find

\begin{align*} \left \Vert \zeta _h\right \Vert _0^2&=a_{u}(\zeta _h,z)=a_{u}(\zeta _h,z-P_hz)+a_u(\zeta _h,P_hz)\\ &=-(k^2+1)(\zeta _h,z-P_hz)-k^2\varepsilon (\vert u \vert ^2\zeta _h,z-P_hz)_{\Omega _{0}}+\mathbf{i} k\langle \zeta _h,z-P_hz\rangle _{\Gamma _{\textrm {imp}}}+a_u(e,P_hz)-a_u(\rho ,P_hz)\\ &=-(k^2+1)(\zeta _h,z-P_hz)-k^2\varepsilon (\vert u \vert ^2\zeta _h,z-P_hz)_{\Omega _{0}}+\mathbf{i} k\langle \zeta _h,z-P_hz\rangle _{\Gamma _{\textrm {imp}}} +J(u_h,P_hz)\\ &\quad +k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,P_hz-z+z\big )+a_u(\rho ,z-P_hz)-(\rho ,\zeta _h)\\ &\lesssim (k^2+k\theta _0) \left \Vert \zeta _h\right \Vert _{0}\left \Vert z-P_hz\right \Vert _{0}+k\left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}}\left \Vert z-P_hz\right \Vert _{0,\Gamma _{\textrm {imp}}} +|\gamma |\eta _{\mathcal{T}} |P_hz|_J \\ &\quad +k\theta _0(\!\left \Vert \zeta _h\right \Vert _{0}+\Vert \rho \Vert _{0})(\!\left \Vert z-P_hz\right \Vert _{0}+k^{-1}\left \Vert \zeta _h\right \Vert _{0}\!)+\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert \left\Vert {\hskip -4.1pt}\left\vert z-P_hz \right\vert {\hskip -4.1pt}\right\Vert +\Vert \rho \Vert _{0}\left \Vert \zeta _h\right \Vert _{0}. \end{align*}

From (2.19), (3.12), (3.14), $kh_{\mathcal{T}}^\alpha \lesssim 1$ , and letting $\theta _0$ sufficiently small, we deduce that

\begin{align*} \left \Vert \zeta _h\right \Vert _0^2&\lesssim \Big (k^2h_{\mathcal{T}}^\alpha \left \Vert \zeta _h\right \Vert _{0}+kh_{\mathcal{T}}^{\frac {\alpha }2}\big ((1+kh_{\mathcal{T}}^{\frac {\alpha }2})\left \Vert \zeta _h\right \Vert _{0}+h_{\mathcal{T}}^{\frac {\alpha }2}\Vert \rho \Vert _{1}\big ) +|\gamma |\eta _{\mathcal{T}} +\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert \Big )\left \Vert z-P_hz\right \Vert _{1} \\ &\quad +|\gamma |\eta _{\mathcal{T}}(\|P_hz-\mathcal I_hz\|_1+|\mathcal I_h z|_J) +h^\alpha _{\mathcal{T}}\Vert \rho \Vert _{1}\left \Vert \zeta _h\right \Vert _{0}\\ &\lesssim (kh_{\mathcal{T}}^{\frac {\alpha }2}+k^2h_{\mathcal{T}}^{\alpha })\big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha \big )\left \Vert \zeta _h\right \Vert _{0}^2 +|\gamma |\eta _{\mathcal{T}} \big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha \big )\left \Vert \zeta _h\right \Vert _{0} \\ &\quad +\big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha +h_{\mathcal{T}}^\alpha \big )\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert \left \Vert \zeta _h\right \Vert _{0}\\ &\lesssim \big (k^2h_{\mathcal{T}}^{1+\frac {\alpha }2}+k^3h_{\mathcal{T}}^{1+\alpha }+(kh_{\mathcal{T}})^\alpha k^{\frac 12}h^{\frac {\alpha }2}+k^{\frac 32+\alpha }h_{\mathcal{T}}^{2\alpha }\big )\left \Vert \zeta _h\right \Vert _{0}^2+\big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha \big )(\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +|\gamma |\eta _{\mathcal{T}})\left \Vert \zeta _h\right \Vert _{0}\\ &\lesssim t(k,h_{\mathcal{T}})\left \Vert \zeta _h\right \Vert _{0}^2+\big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha \big )(\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +|\gamma |\eta _{\mathcal{T}})\left \Vert \zeta _h\right \Vert _{0}. \end{align*}

Therefore, if $k^3h_{\mathcal{T}}^{1+\alpha }$ is sufficiently small, we have

(3.15) \begin{align} \left \Vert \zeta _h\right \Vert _{0}\lesssim \big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha \big )\big (\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +|\gamma |\eta _{\mathcal{T}}\big ). \end{align}

Altogether with (3.11), (3.12), $k^3h_{\mathcal{T}}^{1+\alpha }\leq C_0$ , and $k^{\frac 12}(kh_{\mathcal{T}})^{\alpha }\lesssim 1+k^2h_{\mathcal{T}}$ , we get

(3.16) \begin{align} \left \Vert \nabla \zeta _h\right \Vert _{0}&\lesssim \big (k^{\frac 12}h_{\mathcal{T}}^{\frac {\alpha }2}+k^{\frac 12}(kh_{\mathcal{T}})^{\alpha }+k^2h_{\mathcal{T}}\big )\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +\big ( 1+k^2h_{\mathcal{T}}+k^{\frac 12}(kh_{\mathcal{T}})^\alpha \big )|\gamma |\eta _{\mathcal{T}}\notag \\ &\lesssim (1+k^2h_{\mathcal{T}})(\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +|\gamma |\eta _{\mathcal{T}}) , \end{align}

(3.17) \begin{align} \left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}}&\lesssim \big (kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha +k^2h_{\mathcal{T}}^{1+\frac {\alpha }2}+k^{\frac 12+\alpha }h_{\mathcal{T}}^{\frac {3\alpha }2}\big )\big (\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +|\gamma |\eta _{\mathcal{T}}\big ) +h_{\mathcal{T}}^{\frac {\alpha }2}\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert \notag \\[-3pt] &\quad +k^{-\frac 12}|\gamma |\eta _{\mathcal{T}} +|\gamma |^{\frac 12}\eta _{\mathcal{T}}^{\frac 12} \big ((kh_{\mathcal{T}})^{\frac {1}{2}}+k^{-\frac 14}(kh_{\mathcal{T}})^{\frac {\alpha }2}\big )\big (\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert ^{\frac {1}{2}} +|\gamma |^{\frac {1}{2}}\eta _{\mathcal{T}}^{\frac {1}{2}}\big )\notag \\ &\lesssim \big (h^{\frac {\alpha }2}_{\mathcal{T}}+(kh_{\mathcal{T}})^{\frac 12}\big )\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +\big (k^{-\frac 12} +(kh_{\mathcal{T}})^{\frac 12} \big )|\gamma |\eta _{\mathcal{T}} . \end{align}

Second, we prove (3.6). Using (2.18)–(2.20), (3.8) with $\phi =\rho$ and $v_h=\mathcal I_h\rho$ , (3.2), (3.15), (3.17), $\left \vert \gamma \right \vert \lesssim 1$ , $kh_{\mathcal{T}}^\alpha \lesssim 1$ , and $\left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert \lesssim \|\rho \|_1$ , we derive that

\begin{align*} \Vert \rho \Vert _{1}^2&= b(\rho ,\rho )=b(e,\rho )=a_{u}(e,\rho )+\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u \vert ^2)e,\rho \big )-\mathbf{i} k\langle e,\rho \rangle _{\Gamma _{\textrm {imp}}}\\ &\lesssim \eta _{\mathcal{T}}\Vert \nabla \rho \Vert _{0}+k^2\left \Vert e\right \Vert _{0}\Vert \rho \Vert _{0}+k\left \Vert e\right \Vert _{0,\Gamma _{\textrm {imp}}}\Vert \rho \Vert _{0,\Gamma _{\textrm {imp}}}\\ &\lesssim \big (\eta _{\mathcal{T}}+k^2h_{\mathcal{T}}^\alpha (\!\left \Vert \zeta _h\right \Vert _{0}+\Vert \rho \Vert _{0})+kh^{\frac {\alpha }2}_{\mathcal{T}}(\!\left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}}+\Vert \rho \Vert _{0,\Gamma _{\textrm {imp}}})\big )\Vert \rho \Vert _{1}\\ &\lesssim \big ( 1 + k^3h_{\mathcal{T}}^{1+\alpha } + k^{\frac 32+\alpha }h_{\mathcal{T}}^{2\alpha } + k^{\frac 12}h_{\mathcal{T}}^{\frac {\alpha }2} + k^{\frac 32}h_{\mathcal{T}}^{\frac {1+\alpha }2} \big )\eta _{\mathcal{T}}\|\rho \|_1\\ & \quad + \big ( k^3h_{\mathcal{T}}^{1+\alpha } + k^{\frac 32+\alpha }h_{\mathcal{T}}^{2\alpha } + k^{\frac 32}h_{\mathcal{T}}^{\frac {1+\alpha }2} + k^2h_{\mathcal{T}}^{2\alpha } + kh_{\mathcal{T}}^\alpha \big )\|\rho \|_1^2 \\ &\lesssim \big (1+t(k,h_{\mathcal{T}})\big )\eta _{\mathcal{T}}\Vert \rho \Vert _{1} +t(k,h_{\mathcal{T}})\Vert \rho \Vert _{1}^2, \end{align*}

which implies $\Vert \rho \Vert _{1}\lesssim \eta _{\mathcal{T}}$ when $k^3h_{\mathcal{T}}^{1+\alpha }$ is sufficiently small.

Third, (3.3)–(3.5) comes from (3.15)–(3.17), the triangle inequality, and $k^3h_{\mathcal{T}}^{1+\alpha }\leq C_0$ . Indeed,

\begin{align*} \left \Vert e\right \Vert _{0,\Gamma _{\textrm {imp}}}&\leq \Vert \rho \Vert _{0,\Gamma _{\textrm {imp}}}+\left \Vert \zeta _h\right \Vert _{0,\Gamma _{\textrm {imp}}}\\ &\lesssim \big (h^{\frac {\alpha }2}_{\mathcal{T}} +(kh_{\mathcal{T}})^{\frac 12} \big ) \left\Vert {\hskip -4.1pt}\left\vert \rho \right\vert {\hskip -4.1pt}\right\Vert +\big (k^{-\frac 12} +(kh_{\mathcal{T}})^{\frac 12} \big ) |\gamma |\eta _{\mathcal{T}} \\ &\lesssim \big (h_{\mathcal{T}}^{\frac {\alpha }2}+ k^{\frac 12}h_{\mathcal{T}}^{\frac 12}+(k^{-\frac 12} +k^{\frac 12}h_{\mathcal{T}}^{\frac 12})|\gamma | \big )\eta _{\mathcal{T}}. \end{align*}

Finally, the proof of (3.7) is standard (see, e.g. [Reference Morin, Nochetto and Siebert43]) and is omitted here. This proof is completed.

Remark 3.5. In this proof, we first estimate $P_hu-u_h$ instead of directly estimating $u-u_h$ . When $\gamma =0$ , the upper bound of the FEM for the error on the boundary (3.5) is sharper than the estimate [Reference Duan and Wu21 , (3.10)] for the linear case. However, when $|\gamma |\eqsim 1$ , the upper bound (3.5) becomes $\left \Vert u-u_h\right \Vert _{0,\Gamma _{\mathrm {imp}}}\lesssim (k^{-\frac 12} +k^{\frac 12}h_{\mathcal{T}}^{\frac 12} )\eta _{\mathcal{T}}$ .

Proof. There exists a nonnegative bubble function $\varphi _T$ (see, e.g., [Reference Ainsworth and Oden1]) whose support is $T$ satisfying

(3.21) \begin{align} \left \Vert {\overline {\overline {R}}}_T\right \Vert _{0,T}^2\eqsim \int _T \varphi _T\left \vert {\overline {\overline { R}}}_T \right \vert ^2 \quad \mbox{and}\quad \left \Vert \varphi _T{\overline {\overline {R}}}_T\right \Vert _{0,T}+h_T\left \vert \varphi _T{\overline {\overline { R}}}_T\right \vert _{1,T}\lesssim \left \Vert {\overline {\overline {R}}}_T\right \Vert _{0,T}. \end{align}

Taking $\phi =\varphi _T{\overline {\overline {R}}}_T, v_h=0$ in (3.8) and using the properties (3.21), we have

\begin{align*} \begin{aligned} \left \Vert {\overline {\overline { R}}}_T\right \Vert ^2_{0,T}&\lesssim ({\overline {\overline { R}}}_T,\phi )_T=a_{u}(u-u_h,\phi )-k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,\phi \big )_{\Omega _{0}}+({\overline {\overline {R}}}_T-R_T,\phi )_T\\ &=\big (\nabla (u-P_hu),\nabla \phi \big )-k^2\big ((1+\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u\vert ^2)(u-u_h),\phi \big )\\ &\quad -k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,\phi \big )_{\Omega _{0}}+({\overline {\overline {R}}}_T-R_T,\phi )_T\\ &\lesssim \left \Vert u-P_hu\right \Vert _{1,T}h_T^{-1}\left \Vert {\overline {\overline { R}}}_T\right \Vert _{0,T}+k^2\left \Vert u-u_h\right \Vert _{0,T}\left \Vert {\overline {\overline {R}}}_T\right \Vert _{0,T}\\ &\quad +\left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert k\varepsilon \widehat {M}(f,g)^2\left \Vert u-u_h\right \Vert _{0,T}\left \Vert {\overline {\overline {R}}}_T\right \Vert _{0,T}+ \left \Vert {\overline {\overline { R}}}_T-R_T\right \Vert _{0,T}\left \Vert {\overline {\overline {R}}}_T\right \Vert _{0,T}, \end{aligned} \end{align*}

where we have used $(\nabla (u_h-P_hu),\nabla \phi )=0$ by integration by parts in the second equality and then we get

\begin{align*} h_T\left \Vert {\overline {\overline {R}}}_T\right \Vert _{0,T}\lesssim \left \Vert u-P_hu\right \Vert _{1,T}+k^2h_T\left \Vert u-u_h\right \Vert _{0,T}+h_T\left \Vert {\overline {\overline {R}}}_T-R_T\right \Vert _{0,T}\!. \end{align*}

It follows from (3.4) and the triangle inequality that

(3.22) \begin{align} \begin{aligned} \Big (\!\sum _{T\in \mathcal{T}}h^2_{T}\left \Vert R_{T}\right \Vert ^2_{0,T}\!\Big )^{\frac 12}\lesssim & \left \Vert u-P_hu\right \Vert _1+k^2h_{\mathcal{T}}(kh_{\mathcal{T}}+k^{-\frac 12}(kh_{\mathcal{T}})^\alpha )\eta _{\mathcal{T}}+\Big (\!\sum _{T\in \mathcal{T}}h^2_{T}\left \Vert {\overline {\overline {R}}}_{T}-R_{T}\right \Vert ^2_{0,T}\Big )^{\frac 12}. \end{aligned} \end{align}

Furthermore, a nonnegative edge bubble function $\chi _e$ whose support is $\Omega _e$ satisfies

(3.23) \begin{align} \int _e\chi _e\vert {\overline {\overline { R}}}_e\vert ^2\eqsim \Vert {\overline {\overline { R}}}_e\Vert _{0,e}^2\quad \mbox{and}\quad h_T^{-\frac 12}\Vert \chi _e{\overline {\overline { R}}}_e\Vert _{0,T}+h_T^{\frac 12}\vert \chi _e{\overline {\overline { R}}}_e\vert _{1,T}\lesssim \Vert {\overline {\overline {R}}}_e\Vert _{0,e}\quad \forall T\subset \Omega _e. \end{align}

Denoting by $\phi _1=\sum \limits _{e\in \mathcal{E}}h_e\chi _e{\overline {\overline {R}}}_e$ , then taking $\phi =\phi _1-P_h\phi _1, v_h=-P_h\phi _1$ into (3.8) and using (3.23) and (3.2), we have

\begin{align*} \begin{aligned} \sum _{T\in \mathcal{T}}&\sum _{e\in \partial T}\langle R_e,\phi _1\rangle _e=a_{u}(u-u_h,\phi _1-P_h\phi _1)-\sum _{T\in \mathcal{T}}\Big (R_T,\sum _{e\in {\partial T}}h_e\chi _e{\overline {\overline {R}}}_e\Big )_{T} +J(u_h,P_h\phi _1)\\ &\qquad \qquad \qquad -k^2\varepsilon \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,\phi _1-P_h\phi _1\big )_{\Omega _{0}}\\ &= b(u-P_hu,\phi _1-P_h\phi _1)-\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u\vert ^2)(u-u_h),\phi _1-P_h\phi _1\big ) +J(u_h,P_h\phi _1)\\ & \quad +\mathbf{i} k\langle u-u_h,\phi _1-P_h\phi _1\rangle _{\Gamma _{\textrm {imp}}}-\sum _{T\in \mathcal{T}}\Big (R_T,\sum _{e\in \partial T}h_e\chi _e{\overline {\overline {R}}}_e\Big )_T-k^2\varepsilon \big ((\vert u\vert ^2-\vert u_h\vert ^2)u_h,\phi _1-P_h\phi _1\big )_{\Omega _{0}} \\ &\lesssim |b(u-P_hu,\phi _1)|+k^2\left \Vert u-u_h\right \Vert _0h^\alpha _{\mathcal{T}}\left \Vert \phi _1\right \Vert _1 +|\gamma |\eta _{\mathcal{T}}\|\phi _1\|_1 + k\Vert u-u_h\Vert _{0,\Gamma _{\textrm {imp}}}h^{\frac \alpha 2}_{\mathcal{T}}\Vert \phi _1\Vert _1 \\ &\quad +\Big (\!\sum _{T\in \mathcal{T}}h_T^2\Vert R_T\Vert ^2_{0,T}\Big )^{\frac 12}\Big (\!\sum _{T\in \mathcal{T}}h_T\Vert {\overline {\overline {R}}}_{\partial T}\Vert ^2_{0,\partial T}\Big )^{\frac 12}+k\left \Vert u-u_h\right \Vert _{0}h^\alpha _{\mathcal{T}}\Vert \phi _1\Vert _1\\ &\lesssim \Big (\Vert u-P_hu\Vert _1+k^2h^\alpha _{\mathcal{T}}\left \Vert u-u_h\right \Vert _{0} +|\gamma |\eta _{\mathcal{T}} +kh^{\frac \alpha 2}_{\mathcal{T}}\Vert u-u_h\Vert _{0,\Gamma _{\textrm {imp}}} \\ &\quad +\Big (\!\sum _{T\in \mathcal{T}}h_T^2\Vert R_T\Vert ^2_{0,T}\Big )^{\frac 12}\Big )\Big (\!\sum _{T\in \mathcal{T}}h_T\Vert {\overline {\overline {R}}}_{\partial T}\Vert ^2_{0,{\partial T}}\Big )^{\frac 12}, \end{aligned} \end{align*}

where we have used $\Vert \phi _1\Vert ^2_1\lesssim \sum _{e\in \mathcal{E}}h_e^2\big\Vert \chi _e {\overline {\overline { R}}}_e\big\Vert _{1,\Omega _e}^2 \lesssim \sum _{T\in \mathcal{T}}h_T\Vert {\overline {\overline {R}}}_{\partial T}\Vert ^2_{0,{\partial T}}$ in the last inequality. Then from (3.4) and (3.5), we can deduce that

\begin{align*} \begin{aligned} \sum \limits _{T\in \mathcal{T}}h_T\left \Vert {\overline {\overline {R}}}_{\partial T}\right \Vert ^2_{0,\partial T}&\eqsim \sum \limits _{T\in \mathcal{T}}\sum \limits _{e\in \partial T}\langle {\overline {\overline { R}}}_e,\phi _1\rangle _e=\sum \limits _{T\in \mathcal{T}}\sum \limits _{e\in \partial T}\big (\langle R_e,\phi _1\rangle _e+\langle {\overline {\overline {R}}}_e-R_e,\phi _1\rangle _e\big )\\ &\lesssim \Big (\| u-P_hu \|_1+\big (t(k,h_{\mathcal{T}}) +|\gamma | \big )\eta _{\mathcal{T}}+\Big (\!\sum _{T\in \mathcal{T}}h_T^2\Vert R_T\Vert ^2_{0,T}\Big )^{\frac 12}\Big )\Big (\!\sum _{T\in \mathcal{T}}h_T\Vert {\overline {\overline {R}}}_{\partial T}\Vert ^2_{0,{\partial T}}\Big )^{\frac 12}\\ &\quad +\Big (\!\sum _{T\in \mathcal{T}}h_T\Vert {\overline {\overline {R}}}_{\partial T}-R_{\partial T}\Vert ^2_{0,{\partial T}}\Big )^{\frac 12}\Big (\!\sum _{T\in \mathcal{T}}h_T\Vert {\overline {\overline {R}}}_{\partial T}\Vert ^2_{0,{\partial T}}\Big )^{\frac 12}. \end{aligned} \end{align*}

By taking the estimate (3.22) into the above inequation, we can derive that

(3.24) \begin{align} \sum \limits _{T\in \mathcal{T}}h_T\Vert R_{\partial T}\Vert ^2_{0,{\partial T}}\lesssim \Vert u-P_hu\Vert _1^2+t^2(k,h_{\mathcal{T}})\eta ^2_{\mathcal{T}} +|\gamma |^2\eta ^2_{\mathcal{T}} +\textrm {osc}^2_{\mathcal{T}}. \end{align}

Combining (3.22) and (3.24), we obtain $\eta _{\mathcal{T}}^2\lesssim \Vert u-P_hu\Vert ^2_1+t^2(k,h_{\mathcal{T}})\eta _{\mathcal{T}}^2 +|\gamma |^2\eta ^2_{\mathcal{T}} +\textrm {osc}^2_{\mathcal{T}}$ , which implies (3.18) if $k^3h^{1+\alpha }_{\mathcal{T}}$ and $|\gamma |$ are sufficiently small.

The local lower bound (3.20) can be derived similarly by taking $\phi =\varphi _T\overline {\overline {R_T}}, v_h=0$ and $\phi =\sum _{e\in \partial T}h_e\chi _e\overline {\overline {R_e}},$ $v_h=0$ in (3.8), respectively.

On the other hand, from the standard estimate of the a posteriori lower bound (see, e.g., [Reference Morin, Nochetto and Siebert43]), we can obtain

(3.25) \begin{align} \widetilde {\eta }_{\mathcal{T}}^2\lesssim \Vert u-P_hu\Vert ^2_1+\widetilde {\textrm {osc}}^2_{\mathcal{T}}. \end{align}

To prove (3.19), it remains to estimate the difference between $\textrm {osc}_T$ and $\widetilde {\textrm {osc}}_T$ . Similar to (2.29) and (2.30), we have $\left \Vert P_hu\right \Vert _{L^{\infty }(\Omega _{0})}\lesssim \left \vert \ln h_{\mathcal{T},\textrm {min}} \right \vert ^{\frac 12} k^{-\frac 12} \widehat {M}(f,g)$ if $k^3h_{\mathcal{T}}^2\lesssim 1$ . Therefore,

\begin{align*} \begin{aligned} \!\!\vert \widetilde {\textrm {osc}}_T&-\textrm {osc}_T\vert ^2\leq h^2_{T}\big \Vert \widetilde R_{T}-R_{T}-\overline {\overline {\widetilde {R}_{T}-R_{T}}}\big \Vert ^2_{0,T}+h_T\big \Vert \widetilde {R}_{\partial T}-R_{\partial T}-\overline {\overline {\widetilde {R}_{\partial T}-R_{\partial T}}}\big \Vert ^2_{0,{\partial T}}\\ &= h^2_{T}\big \Vert \big (k^2(1+\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u \vert ^2)+1\big )u-\overline {\overline {\big (k^2(1+\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u \vert ^2)+1\big )u}}\big \Vert _{0,T}+h_T\Vert \mathbf{i} k(u-\overline {\overline {u}})\Vert _{0,\partial T\cap \Gamma _{\textrm {imp}}}\\ &\leq h^2_{T}\big \Vert (k^2+1)(u-P_hu)+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\big ((\vert u\vert ^2-\vert P_hu\vert ^2)u+\vert P_hu\vert ^2(u-P_hu)\big )\big \Vert ^2_{0,T}\!+h_T\Vert \mathbf{i} k(u-P_hu)\Vert ^2_{0,{\partial T}\cap \Gamma _{\textrm {imp}}}\\ &\lesssim \big (k^4h^2_T\Vert u-P_hu\Vert _{0,T}^2+k^2h_T\Vert u-P_hu\Vert ^2_{0,{\partial T}\cap \Gamma _{\textrm {imp}}}\big ). \end{aligned} \end{align*}

From the definitions of $\textrm {osc}_{\mathcal{T}}$ and $\widetilde {\textrm {osc}}_{\mathcal{T}}$ , (3.6), $k^3h_{\mathcal{T}}^{1+\alpha }\leq C_0$ and trace inequality, we have

(3.26) \begin{align} \begin{aligned} \vert \widetilde {\textrm {osc}}_{\mathcal{T}}-\textrm {osc}_{\mathcal{T}}\vert ^2&\leq \sum \limits _{T\in \mathcal{T}}\vert \widetilde {\textrm {osc}}_T-\textrm {osc}_T\vert ^2\lesssim k^4h_{\mathcal{T}}^2\left \Vert u-P_hu\right \Vert _{0}^2+k^2h_{\mathcal{T}}\left \Vert u-P_hu\right \Vert _{0,\Gamma _{\textrm {imp}}}^2\\ &\lesssim (k^4h_{\mathcal{T}}^2 h_{\mathcal{T}}^{2\alpha }+k^2h_{\mathcal{T}} h_{\mathcal{T}}^{\alpha })\eta _{\mathcal{T}}^2\lesssim \eta _{\mathcal{T}}^2. \end{aligned} \end{align}

Then together with (3.25) and (3.18), (3.19) is derived.

Remark 3.7. (i) Combining (3.6) with (3.18), we get $\eta _{\mathcal{T}}\eqsim \left \Vert u-P_hu\right \Vert _{1}+\mathrm {osc}_{\mathcal{T}}$ . That is to say, the error estimator of $u_h$ fits the error of $P_hu$ well and underestimates seriously the error of $u_h$ in the preasymptotic regime. This behaviour is first observed by Babuška et al. [ Reference Babuška, Ihlenburg, Strouboulis and Gangaraj3 ] for a one-dimensional linear problem. We prove this observation for the first time for the NLH problem.

(ii) From (3.26), there exists a constant $C_{\mathrm {osc}}$ such that for any $\mathcal{M}\subset \mathcal{T}$ , the following estimate holds

(3.27) \begin{align} \vert \mathrm {osc}_{\mathcal{M}}-\widetilde {\mathrm {osc}}_{\mathcal{M}}\vert ^2\leq C_{\mathrm {osc}} (k^4h_{\mathcal{T}}^2 h_{\mathcal{T}}^{2\alpha }+k^2h_{\mathcal{T}} h_{\mathcal{T}}^{\alpha })\eta _{\mathcal{T}}^2. \end{align}

This estimate will be used twice subsequently.

Proof. Similar to (3.8), for any $\phi \in H^1_{\Gamma _{\textrm {Dir}}}(\Omega )$ , we have

\begin{align*} b(u-P_hu,\phi )=\sum \limits _{T\in \mathcal{T}}\big ((\widetilde {R}_T,\phi )_T+\langle \widetilde {R}_{\partial T},\phi \rangle _{\partial T}\big ). \end{align*}

On the other hand, it follows from the definitions of $a_u(\cdot ,\cdot )$ , $b(\cdot ,\cdot )$ and (2.18):

\begin{align*} &b(u-P_hu,\phi )=b(u-u_h,\phi -P_h\phi )\\ & \quad =a_{u}(u-u_h,\phi -P_h\phi )+\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u \vert ^2)(u-u_h),\phi -P_h\phi \big )-\mathbf{i} k\langle u-u_h,\phi -P_h\phi \rangle _{\Gamma _{\textrm {imp}}}. \end{align*}

By combining (3.8) with the above two identities, we obtain

(3.30) \begin{align} &\sum _{T\in \mathcal{T}}\big ((R_T,\phi )_T+\langle R_{\partial T},\phi \rangle _{\partial T}\big )=\sum \limits _{T\in \mathcal{T}}\big ((\widetilde {R}_T,\phi )_T+\langle \widetilde {R}_{\partial T},\phi \rangle _{\partial T}\big )-k^2\varepsilon \big ((\vert u\vert ^2-\vert u_h\vert ^2)u_h,\phi -P_h\phi \big )_{\Omega _{0}}\nonumber\\&\quad +J(u_h,P_h\phi ) -\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u\vert ^2)(u-u_h),\phi -P_h\phi \big )+\mathbf{i} k\langle u-u_h,\phi - P_h\phi \rangle _{\Gamma _{\textrm {imp}}}. \end{align}

Denote by $\widetilde {\mathcal{M}}\,:\!=\,\bigcup _{T\in \mathcal{M}}\Sigma _{\mathcal{T}}(T)$ . Taking $\phi =\sum \limits _{T\in \widetilde {\mathcal{M}}}h^2_{T}\varphi _{T}{\overline {\overline {R}}}_{T}$ in (3.30), we get

\begin{align*} \begin{aligned} \sum _{T\in \widetilde {\mathcal{M}}}(R_{T},\phi )_{T}&=\sum _{T\in \widetilde {\mathcal{M}}}(\widetilde R_{T},\phi )_{T}-k^2\varepsilon \big ((\vert u\vert ^2-\vert u_h\vert ^2)u_h,\phi -P_h\phi \big )_{\Omega _{0}} +J(u_h,P_h\phi )\\ &\quad -\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u\vert ^2)(u-u_h),\phi -P_h\phi \big )+\mathbf{i} k\langle u-u_h,\phi -P_h\phi \rangle _{\Gamma _{\textrm {imp}}}. \end{aligned} \end{align*}

Then, it follows from (3.21) and (2.19) that

\begin{align*} \begin{aligned} &\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert {\overline {\overline {R}}}_{T}\Vert ^2_{0,T}\lesssim \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}({\overline {\overline { R}}}_{T},\varphi _{T}{\overline {\overline { R}}}_{T})_{T}=\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}(R_{T},\varphi _{T}{\overline {\overline { R}}}_{T})_{T}+\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}({\overline {\overline { R}}}_{T}-R_{T},\varphi _{T}{\overline {\overline {R}}}_{T})_{T}\\ &\quad =\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}(\widetilde {R}_{T},\varphi _{T}{\overline {\overline { R}}}_{T})_{T}+\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}({\overline {\overline { R}}}_{T}-R_{T},\varphi _{T}{\overline {\overline {R}}}_{T})_{T} +J(u_h,P_h\phi ) +\mathbf{i} k\langle u-u_h,\phi -P_h\phi \rangle _{\Gamma _{\textrm {imp}}}\\ & \quad -k^2\varepsilon \big ((\vert u\vert ^2-\vert u_h\vert ^2)u_h,\phi -P_h\phi \big )_{\Omega _{0}}-\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u\vert ^2)(u-u_h),\phi -P_h\phi \big )\\& \quad \lesssim \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert \widetilde {R}_{T}\Vert _{0,T}\Vert {\overline {\overline { R}}}_{T}\Vert _{0,T}+\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert R_{T}-{\overline {\overline { R}}}_{T}\Vert _{0,T}\Vert {\overline {\overline {R}}}_{T}\Vert _{0,T} +|\gamma |\eta _{\mathcal{T}}\|\phi \|_1\\ & \quad +k\Vert u-u_h\Vert _{0,\Gamma _{\textrm {imp}}}h^{\frac \alpha 2}_{\mathcal{T}}\Vert \phi \Vert _1+k\left \Vert u-u_h\right \Vert _{0}h_{\mathcal{T}}^{\alpha }\Vert \phi \Vert _1+k^2\Vert u-u_h\Vert _0h_{\mathcal{T}}^{\alpha }\Vert \phi \Vert _1\\& \quad \lesssim \bigg (\Big (\!\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert \widetilde {R}_{T}\Vert ^2_{0,T}\Big )^{\frac 12}+\big (t(k,h_{\mathcal{T}}) +|\gamma |\big )\eta _{\mathcal{T}}+\Big (\!\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert R_{T}-{\overline {\overline { R}}}_{T}\Vert _{0,T}^2\Big )^{\frac 12}\bigg )\Big (\!\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert {\overline {\overline { R}}}_{T}\Vert ^2_{0,T}\Big )^{\frac 12}, \end{aligned} \end{align*}

where we have used $\Vert \phi \Vert ^2_1\lesssim \sum _{T\in \widetilde {\mathcal{M}}}h^4_{T}\Vert \varphi _T{\overline {\overline { R}}}_T\Vert ^2_{1,T}\lesssim \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert {\overline {\overline {R}}}_T\Vert ^2_{0,T}$ , (3.2), the upper bounds (3.4) and (3.5), and (2.22) to derive the last inequality. Then, we proceed

(3.31) \begin{align} \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert R_{T}\Vert ^2_{0,T}\lesssim \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert \widetilde {R}_{T}\Vert ^2_{0,T}+\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta ^2_{\mathcal{T}}+\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert R_{T}-{\overline {\overline {R}}}_{T}\Vert _{0,T}^2. \end{align}

Denote $\mathcal{E}_{\mathcal{M}}$ the set of all edges in $\mathcal{M}$ and let $\phi =\sum _{e\in \mathcal{E}_{\mathcal{M}}}h_e\chi _e {\overline {\overline {R}}}_e$ in (3.30). From (3.23) and (2.19), we have

\begin{align*} \begin{aligned} \bigg |\sum _{T\in \widetilde {\mathcal{M}}}\sum _{\substack {e\subset \partial T\\e\in \mathcal{E}_{\mathcal{M}}}}\langle R_e,\phi \rangle _e\bigg |&= \bigg |-\sum _{T\in \widetilde {\mathcal{M}}}(R_T,\phi )_T + \sum \limits _{T\in \widetilde {\mathcal{M}}}\big ((\widetilde {R}_T,\phi )_T+\langle \widetilde {R}_{\partial T},\phi \rangle _{\partial T}\big ) +J(u_h,P_h\phi ) +\mathbf{i} k\langle u-u_h,\phi -P_h\phi \rangle _{\Gamma _{\textrm {imp}}}\\ &\quad -k^2\varepsilon \big ((\vert u\vert ^2-\vert u_h\vert ^2)u_h,\phi -P_h\phi \big )_{\Omega _{0}}-\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u\vert ^2)(u-u_h),\phi -P_h\phi \big )\bigg |\\ &\lesssim \Big (\!\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert R_{T}\Vert ^2_{0,T}+\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert \widetilde R_{T}\Vert ^2_{0,T}+\sum _{T\in \mathcal{M}}h_T\Vert \widetilde R_{\partial T}\Vert ^2_{0,{\partial T}}\Big )^{\frac 12}\Big (\!\sum _{T\in \mathcal{M}}h_T\Vert {\overline {\overline {R}}}_{\partial T}\Vert ^2_{0,{\partial T}}\Big )^{\frac 12}\\ &\quad +\big ( |\gamma |\eta _{\mathcal{T}} + kh^{\frac \alpha 2}_{\mathcal{T}}\| u-u_h \|_{0,\Gamma _{\textrm {imp}}}+k^2h^\alpha _{\mathcal{T}}\| u-u_h \|_0\big )\| \phi \|_1. \end{aligned} \end{align*}

We can easily find that

\begin{equation*}\left \Vert \phi \right \Vert _{1}^2\lesssim \sum _{e\in \mathcal{E}_{\mathcal{M}}}h_e^2\left \Vert \chi _e {\overline {\overline {R}}} _e\right \Vert _{1,\Omega _e}^2\lesssim \sum _{T\in \mathcal{M}} h_T \left \Vert {\overline {\overline {R}}}_{\partial T}\right \Vert _{0,\partial T}^2 ,\end{equation*}

then we deduce from (3.31) that

\begin{align*} \sum _{T\in \mathcal{M}} h_T \left \Vert {\overline {\overline {R}}}_{\partial T}\right \Vert _{0,\partial T}^2&\eqsim \sum _{T\in \widetilde {\mathcal{M}}}\sum _{\substack {e\subset \partial T\\e\in \mathcal{E}_{\mathcal{M}}}}\langle {\overline {\overline {R}}}_e, h_e\chi _e {\overline {\overline {R}}}_e\rangle _e =\sum _{T\in \widetilde {\mathcal{M}}}\sum _{\substack {e\subset \partial T\\e\in \mathcal{E}_{\mathcal{M}}}}\langle R_e, \phi \rangle _e + \sum _{T\in \widetilde {\mathcal{M}}}\sum _{\substack {e\subset \partial T\\e\in \mathcal{E}_{\mathcal{M}}}}\langle {\overline {\overline {R}}}_e-R_e, \phi \rangle _e \\ &\lesssim \Big (\!\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\big (\| \widetilde R_{T} \|_{0,T}^2+\| R_{T}-{\overline {\overline {R}}}_{T} \|^2_{0,T}\big ) + \sum _{T\in \mathcal{M}} h_T\| \widetilde R_{\partial T} \|_{0,\partial T}^2 \\ &\quad +\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta ^2_{\mathcal{T}} + \sum _{T\in \mathcal{M}} h_T\left \Vert R_{\partial T}-{\overline {\overline {R}}}_{\partial T}\right \Vert _{0,\partial T}^2\!\Big )^{\frac 12}\Big (\!\sum _{T\in \mathcal{M}} h_T \left \Vert {\overline {\overline {R}}}_{\partial T}\right \Vert _{0,\partial T}^2\Big )^{\frac 12}. \end{align*}

As a consequence, we have the following estimate with the aid of the triangle inequality

\begin{align*} \sum _{T\in \mathcal{M}} h_T \left \Vert R_{\partial T}\right \Vert _{0,\partial T}^2&\lesssim \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\big (\| \widetilde R_{T} \|_{0,T}^2+\| R_{T}-{\overline {\overline {R}}}_{T} \|^2_{0,T}\big )+ \sum _{T\in \mathcal{M}}h_T\| \widetilde R_{\partial T} \|_{0,\partial T}^2\\ &\quad +\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta ^2_{\mathcal{T}} + \sum _{T\in \mathcal{M}} h_T\left \Vert R_{\partial T}-{\overline {\overline {R}}}_{\partial T}\right \Vert _{0,\partial T}^2. \end{align*}

Altogether with (3.31) and (3.27), it reads

\begin{align*} \begin{aligned} \eta _{\mathcal{M}}^2&\eqsim \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\Vert R_{T}\Vert ^2_{0,T}+\sum _{T\in \mathcal{M}} h_T \left \Vert R_{\partial T}\right \Vert _{0,\partial T}^2\\ &\lesssim \widetilde {\eta }_{\mathcal{M}}^2+\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta _{\mathcal{T}}^2+ \sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\| R_{T}-{\overline {\overline {R}}}_{T} \|^2_{0,T}+\sum _{T\in \mathcal{M}} h_T\left \Vert R_{\partial T}-{\overline {\overline {R}}}_{\partial T}\right \Vert _{0,\partial T}^2\\ &\lesssim \widetilde {\eta }_{\mathcal{M}}^2+\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta _{\mathcal{T}}^2+\widetilde \eta _{\mathcal{M}}^2+(k^4h_{\mathcal{T}}^{2+2\alpha }+k^2h_{\mathcal{T}}^{1+\alpha })\eta _{\mathcal{T}}^2\\ &\lesssim \widetilde {\eta }_{\mathcal{M}}^2+\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta _{\mathcal{T}}^2. \end{aligned} \end{align*}

By the similar argument, we arrive at

\begin{align*} \widetilde \eta _{\mathcal{M}}^2&\lesssim \eta _{\mathcal{M}}^2 +\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta _{\mathcal{T}}^2+\sum _{T\in \widetilde {\mathcal{M}}}h^2_{T}\| \widetilde R_{T}-\overline {\overline {\widetilde R_{T}}} \|^2_{0,T}+\sum _{T\in \mathcal{M}} h_T\left \Vert \widetilde R_{\partial T}-\overline {\overline {\widetilde R_{\partial T}}}\right \Vert _{0,\partial T}^2\\ &\lesssim \eta _{\mathcal{M}}^2 +\big (t^2(k,h_{\mathcal{T}})+|\gamma |^2\big )\eta _{\mathcal{T}}^2. \end{align*}

Then, we get the desired estimates (3.28)–(3.29).

Proof. It follows form (3.28)-(3.29) that

\begin{align*} \widetilde \eta _{\mathcal{M}}\ge &\ \frac {\eta _{\mathcal{M}}}{C_{\textrm {eq}}}-(t(k,h_{\mathcal{T}})+|\gamma |)\eta _{\mathcal{T}} \ge \ \frac {\theta _D\eta _{\mathcal{T}}}{C_{\textrm {eq}}}-(t(k,h_{\mathcal{T}})+|\gamma |)\eta _{\mathcal{T}}=\ \widetilde {\theta }_D\widetilde C_{\textrm {eq}}(1+t(k,h_{\mathcal{T}})+|\gamma |)\eta _{\mathcal{T}}\ge \ \widetilde {\theta }_D\widetilde \eta _{\mathcal{T}}. \end{align*}

Remark 4.1. (i) Unlike the algorithm in [ Reference Duan and Wu21 ] which refines the broader mesh $\widetilde {\mathcal{M}}_n\,:\!=\,\bigcup _{T\in \mathcal{M}_n}\Sigma _{\mathcal{T}_n}(T)$ in order to apply the equivalent relationships of the error estimators to derive the convergence and quasi-optimality of $u_n$ , the above adaptive algorithm is the same as the standard adaptive algorithm for elliptic problems since we modify the definitions of error estimators.

(ii) In step 1, we assume solving (2.14) is accurate but in practical computation we utilize Newton’s iteration until the relative error between two consecutive iterative solutions falls below a prescribed tolerance. Indeed, when using the frozen-nonlinearity iteration with the iteration number fixed, it is named an adaptive Kačanov method for quasi-linear elliptic partial differential equations (see, e.g., [ Reference Garau, Morin and Zuppa22 ]). For our NLH problem, the convergence cannot be theoretically guaranteed. However, the numerical results are promising with an iteration number specified. We will study this convergence in our future work.

Proof. It is from the definitions of $\widetilde \eta _T^*$ , the triangle inequality, the trace inequality and the inverse estimate that

\begin{align*} \big \vert \widetilde \eta _T^*(v_{h1})-\widetilde \eta _T^*(v_{h2})\big \vert ^2&\leq h_{T}^2\left \Vert \widetilde R_T(v_{h1})-\widetilde R_T(v_{h2})\right \Vert _{0,T}^2+h_{T}\left \Vert \widetilde R_{{\partial T}}(v_{h1})-\widetilde R_{{\partial T}}(v_{h2})\right \Vert _{0,{\partial T}}^2\\ &\quad +\frac 1{6}h^2_T\sum _{T'\in \Sigma _{\mathcal{T}}(T)\setminus \{T\}}\left \Vert \widetilde R_{T'}(v_{h1})-\widetilde R_{T'}(v_{h2})\right \Vert _{0,T'}^2 \\ &=h_{T}^2\left \Vert v_{h1}-v_{h2}\right \Vert _{0,T}^2+\frac {h_T}{4}\left \Vert [\![\nabla (v_{h1}-v_{h2})]\!]\right \Vert _{0,{\partial T}\cap \Omega }^2+h_{T}\left \Vert \nabla (v_{h1}-v_{h2})\cdot n\right \Vert _{0,{\partial T}\cap \Gamma _{\textrm {imp}}}^2\\ &\quad +\frac 1{6}h^2_T\sum _{T'\in \Sigma _{\mathcal{T}}(T)\setminus \{T\}}\left \Vert v_{h1}-v_{h2}\right \Vert _{0,T'}^2\\ &\leq C_{\textrm {per}}^2\big (h^2_T\left \Vert v_{h1}-v_{h2}\right \Vert _{0,\Omega _{\mathcal{T}}(T)}^2+\left \Vert \nabla (v_{h1}-v_{h2})\right \Vert _{0,\Omega _{\mathcal{T}}(T)}^2\!\big )\\ &\leq C_{\textrm {per}}^2\left \Vert v_{h1}-v_{h2}\right \Vert _{1,\Omega _{\mathcal{T}}(T)}^2. \end{align*}

Then, the conclusion comes from finding the square root.

Proof. First, it can be directly derived from Lemma 4.2 and the Young’s inequality that

(4.3) \begin{align} \widetilde \eta _{\mathcal{T}_{n+1}}^*(\widetilde u_{n+1})^2\leq (1+\delta )\widetilde \eta _{\mathcal{T}_{n+1}}^*(\widetilde u_n)^2+ 4(1+\delta ^{-1})C_{\textrm {per}}^2\left \Vert \widetilde u_{n+1}-\widetilde u_n\right \Vert _{1}^2. \end{align}

Then for $T\in \mathcal{T}_n,$ we define $T_*\,:\!=\,\{T_1\in \mathcal{T}_{n+1}:\; T_1\subset T\}$ and handle $\widetilde \eta _{\mathcal{T}_{n+1}}^*(\widetilde u_{n})^2$ in two parts:

\begin{align*} \widetilde \eta _{\mathcal{T}_{n+1}}^*(\widetilde u_{n})^2&=\sum _{\substack {T\in \mathcal{M}_n\\ T_1\in T_*}}\widetilde \eta _{T_1}^*(\widetilde u_{n})^2+\sum _{\substack {T\in \mathcal{T}_{n}\setminus \mathcal{M}_n\\ T_1\in T_*}}\widetilde \eta _{T_1}^*(\widetilde u_{n})^2. \end{align*}

Part I: for $T\in \mathcal{M}_n$ , we have

\begin{align*} \sum _{T_1\in T_*}\widetilde \eta _{T_1}^*(\widetilde u_{n})^2&=\sum _{T_1\in T_*}\Bigg (h_{T_1}^2\big \Vert \widetilde R_{T_1}\big \Vert _{0,T_1}^2+h_{T_1}\big \Vert \widetilde R_{\partial T_1}\big \Vert _{0,\partial T_1}^2+\frac 1{6}h^2_{T_1}\sum _{T'\in \Sigma _{\mathcal{T}_{n+1}}(T_1)\setminus \{T_1\}}\big \Vert \widetilde R_{T'}\big \Vert _{0,T'}^2\Bigg )\\ &\leq\sum _{T_1\in T_*}\Bigg (2^{-b}h^2_{T}\big \Vert \widetilde R_{T_1}\big \Vert _{0,T_1}^2+2^{-\frac b2}h_{T}\big \Vert \widetilde R_{\partial T_1}\big \Vert _{0,\partial T_1}^2+\frac {2^{-b}}{6}h_T^2\sum _{\substack {T'\in \Sigma _{\mathcal{T}_{n+1}}(T_1)\setminus \{T_1\}\\ T'\not \subset T}}\big \Vert \widetilde R_{T'}\big \Vert _{0,T'}^2\\ &\quad +\frac {2^{-b}}{6}h_T^2\sum _{\substack {T'\in \Sigma _{\mathcal{T}_{n+1}}(T_1)\setminus \{T_1\}\\ T'\subset T}}\big \Vert \widetilde R_{T'}\big \Vert _{0,T'}^2\Bigg )\\ &\leq 2^{-b}h^2_{T}\big \Vert \widetilde R_T\big \Vert _{0,T}^2+2^{-\frac b2}h_{T}\big \Vert \widetilde R_{{\partial T}}\big \Vert _{0,\partial T}^2+\frac {2^{-b}}{6}h^2_T\sum _{T'\in \Sigma _{\mathcal{T}_n}(T)\setminus \{T\}}\big \Vert \widetilde R_{T'}\big \Vert _{0,T'}^2 +\frac {2^{-b}\cdot 3}{6}h^2_T\big \Vert \widetilde R_T\big \Vert _{0,T}^2\\ &\leq \lambda _b\widetilde \eta ^*_T(\widetilde u_n)^2. \end{align*}

Since $\widetilde \eta ^*_{\mathcal{M}_n}(\widetilde u_n)^2=\sum _{T\in \mathcal{M}_n}\widetilde \eta ^*_T(\widetilde u_n)^2$ , we get

(4.4) \begin{align} \sum _{\substack {T\in \mathcal{M}_n\\ T_1\in T_*}}\widetilde \eta _{T_1}^*(\widetilde u_{n})^2\leq \lambda _b\widetilde \eta ^*_{\mathcal{M}_n}(\widetilde u_n)^2. \end{align}

Part II: let $\mathcal R$ be the set of refined elements in $\mathcal{T}_n$ . Exactly, $\mathcal{M}_n\subset \mathcal R$ . For $T\in \mathcal{T}_n{\setminus}\mathcal R$ , $T_*$ equals $\{T\}$ , and

\begin{align*} \sum _{T_1\in T_*}\widetilde \eta _{T_1}^*(\widetilde u_n)^2\leq \widetilde \eta _{T}^*(\widetilde u_n)^2. \end{align*}

On the other hand, for $T\in (\mathcal{T}_n{\setminus}\mathcal{M}_n)\cap \mathcal R$ , similar to the first estimate in Part I, we derive

\begin{align*} \sum _{T_1\in T_*}\widetilde \eta _{T_1}^*(\widetilde u_n)^2&=\sum _{ T_1\in T_*}\Bigg (h_{T_1}^2\big \Vert \widetilde R_{T_1}\big \Vert _{0,T_1}^2+h_{T_1}\big \Vert \widetilde R_{\partial T_1}\big \Vert _{0,\partial T_1}^2+\frac 1{6}h^2_{T_1}\sum _{T'\in \Sigma _{\mathcal{T}_{n+1}}(T_1)\setminus \{T_1\}}\big \Vert \widetilde R_{T'}\big \Vert _{0,T'}^2\Bigg )\leq \lambda _1\widetilde \eta _{T}^*(\widetilde u_n)^2. \end{align*}

As a conclusion, we can get

(4.5) \begin{align} \sum _{\substack {T\in \mathcal{T}_{n}\setminus \mathcal{M}_n\\ T_1\in T_*}}\widetilde \eta _{T_1}^*(\widetilde u_{n})^2\leq \sum _{T\in \mathcal{T}_{n}\setminus \mathcal{M}_n}\widetilde \eta _{T}^*(\widetilde u_{n})^2 = \widetilde \eta _{\mathcal{T}_n}^*(\widetilde u_n)^2-\widetilde \eta _{\mathcal{M}_n}^*(\widetilde u_n)^2. \end{align}

Then combining (4.4) and (4.5), we have

(4.6) \begin{align} \widetilde \eta _{\mathcal{T}_{n+1}}^*(\widetilde u_{n})^2\leq \widetilde \eta _{\mathcal{T}_n}^*(\widetilde u_n)^2-(1-\lambda _b)\widetilde \eta _{\mathcal{M}_n}^*(\widetilde u_n)^2. \end{align}

Finally, (4.2) comes from (4.3) and (4.6). This completes the proof of the lemma.

Proof. The key point is to prove the contraction property: there exist constants $\beta \gt 0$ and $C_{\textrm {conv}}\in (0,1)$ such that

(4.7) \begin{align} \left \Vert u-\widetilde u_{n+1}\right \Vert _{1}^2+\beta \widetilde \eta _{\mathcal{T}_{n+1}}^*(\widetilde u_{n+1})^2 \leq C_{\textrm {conv}}^2\big (\!\left \Vert u-\widetilde u_n\right \Vert _{1}^2+\beta \widetilde \eta _{\mathcal{T}_n}^*(\widetilde u_n)^2\big ). \end{align}

Combining (2.18) and Lemma 4.3, we choose $\beta =\frac 1{4(1+\delta ^{-1})C_{\textrm {per}}^2}$ to get

\begin{align*} \left \Vert u-\widetilde u_{n+1}\right \Vert _{1}^2+\beta \widetilde \eta _{\mathcal{T}_{n+1}}^*(\widetilde u_{n+1})^2&\leq \left \Vert u-\widetilde u_n\right \Vert _{1}^2-\left \Vert \widetilde u_{n+1}-\widetilde u_n\right \Vert _{1}^2+\beta (1+\delta )\big (\widetilde \eta _{\mathcal{T}_{n}}^*(\widetilde u_{n})^2-(1-\lambda _b) \widetilde \eta _{\mathcal{M}_{n}}^*(\widetilde u_{n})^2\big )\\ &\quad +4\beta (1+\delta ^{-1})C_{\textrm {per}}^2\left \Vert \widetilde u_{n+1}-\widetilde u_n\right \Vert _{1}^2\\ &=\left \Vert u-\widetilde u_n\right \Vert _{1}^2+\beta (1+\delta )\big (\widetilde \eta _{\mathcal{T}_{n}}^*(\widetilde u_{n})^2-(1-\lambda _b) \widetilde \eta _{\mathcal{M}_{n}}^*(\widetilde u_{n})^2\big ). \end{align*}

Noting that $\widetilde \eta _{\mathcal{M}_{n}}(\widetilde u_{n})\geq \widetilde \theta _{\textrm {D}} \widetilde \eta _{\mathcal{T}_n}(\widetilde u_n)$ (see Corollary 3.10) and $\left \Vert u-\widetilde u_n\right \Vert _{1}\leq C_{\textrm {up}}\widetilde \eta _{\mathcal{T}_n}$ for some constant $C_{\textrm {up}}\gt 0$ (see (3.6)) when $k^3h^{1+\alpha }$ and $|\gamma |$ are sufficiently small, we have

\begin{align*} \widetilde \eta ^*_{\mathcal{M}_n}\geq c_1\widetilde \eta _{\mathcal{M}_n}\geq c_1\widetilde \theta _{\textrm {D}}\widetilde \eta _{\mathcal{T}_n}\geq \frac {c_1\widetilde \theta _{\textrm {D}}}{c_2}\widetilde \eta ^*_{\mathcal{T}_n},\quad \left \Vert u-\widetilde u_n\right \Vert _{1}\leq \frac {C_{\textrm {up}}}{c_1}\widetilde \eta _{\mathcal{T}_n}^*. \end{align*}

Then, for any $\mu \in (0,1)$ ,

\begin{align*} \left \Vert u-\widetilde u_{n+1}\right \Vert _{1}^2+\beta \big(\widetilde \eta _{\mathcal{T}_{n+1}}^*\big)^2&\leq \Big (1-\beta (1+\delta )(1-\lambda _b)\mu \Big (\frac {c_1\widetilde \theta _{\textrm {D}}}{c_2C_{\textrm {up}}}\Big )^2\Big )\left \Vert u-\widetilde u_n\right \Vert _{1}^2\\ &\quad +\beta (1+\delta )\Big (1-(1-\lambda _b)(1-\mu )\Big (\frac {c_1\widetilde \theta _{\textrm {D}}}{c_2}\Big )^2\Big )(\widetilde \eta _{\mathcal{T}_{n}}^*)^2. \end{align*}

To obtain (4.7), we let

\begin{align*} C_{\textrm {conv}}^2=\max \Big \{1-\beta (1+\delta )(1-\lambda _b)\mu \Big (\frac {c_1\widetilde \theta _{\textrm {D}}}{c_2C_{\textrm {up}}}\Big )^2, \,(1+\delta )\Big (1-(1-\lambda _b)(1-\mu )\Big (\frac {c_1\widetilde \theta _{\textrm {D}}}{c_2}\Big )^2\Big )\Big \} \end{align*}

and choose $\delta$ sufficiently small such that $C_{\textrm {conv}}\lt 1$ .

Finally, it follows from the upper bounds (3.3) and (3.4), the equivalences of and $\widetilde \eta _{\mathcal{T}}^*$ , (4.7), and the lower bound (3.19) that

\begin{align*} \left\Vert {\hskip -4.1pt}\left\vert u-u_n \right\vert {\hskip -4.1pt}\right\Vert &\lesssim (1+k^{\frac 12}(kh_n)^{\alpha }+k^2h_n) \eta _{\mathcal{T}_n}\lesssim (1+k^2h_n) \widetilde \eta _{\mathcal{T}_n}^*\\ &\lesssim (1+k^2h_n)(\!\left \Vert u-\widetilde u_{n}\right \Vert _{1}^2+ \beta (\widetilde \eta _{\mathcal{T}_n}^*)^2)^{1/2}\\ &\lesssim (1+k^2h_n)(\!\left \Vert u-\widetilde u_{0}\right \Vert _{1}+ \widetilde \eta _{\mathcal{T}_0}^*)C_{\textrm {conv}}^n\\ &\lesssim (1+k^2h_n) (\!\left \Vert u-\widetilde u_{0}\right \Vert _{1}+ \textrm {osc}_{\mathcal{T}_0})C_{\textrm {conv}}^n. \end{align*}

Altogether with

(4.8) \begin{align} &\textrm {osc}_{\mathcal{T}_0} = {\Big (\!\sum _{T\in \mathcal{T}_0} \big (h_T^2\| f-\overline {\overline {f}} \|^2_{0,T}+ h_T\| g-\overline {\overline {g}} \|_{0,\partial T\cap \Gamma _{\textrm {imp}}}^2\big )\Big )}^{\frac {1}{2}}\lesssim h_0 \big (\| f \|_0+\| g \|_{\frac 12, \Gamma _{\textrm {imp}}}\big ),\\ &{\left \Vert u-\widetilde u_0\right \Vert }_{1}\lesssim (kh_0+k^{-\frac 12}(kh_0)^{\alpha })\widehat {M}(f,g),\notag \end{align}

we get the desired convergence result.

Proof. By the orthogonality of $\widetilde u_n$ , we have for any $v_h\in V_{n}$ ,

\begin{align*} \left \Vert u-\widetilde u_n\right \Vert _{1}^2=\left \Vert u-v_h\right \Vert _{1}^2-\left \Vert \widetilde u_n-v_h\right \Vert _{1}^2\leq \left \Vert u-v_h\right \Vert _{1}^2. \end{align*}

Moreover, we see from the definition of $\widetilde {\textrm {osc}}$ that $\widetilde {\textrm {osc}}_{\mathcal{T}_n}(v_h)$ is independent of $v_h\in V_{n}$ , and hence (4.9) is derived.

Proof. Let $\Omega _{\mathcal R}=\cup \{T\,:\,T\in \mathcal R\}$ be the union of refined elements, and denote by $\Omega _i, i=1,\ldots ,I$ , the connected components of its interior. We denote $\mathcal{T}^i=\{T\in \mathcal{T}\,:\, T\subset \bar \Omega _i\}$ and $V_{\mathcal{T}^i}$ be the restriction of $V_{\mathcal{T}}$ to $\Omega _i$ . For the error $E_*\,:\!=\,\widetilde u_{\mathcal{T}_*}-\widetilde u_{\mathcal{T}}$ , we construct $\Pi _hE_*\in V_{\mathcal{T}}$ by

\begin{align*} \Pi _hE_*\,:\!=\,\Pi _{hi} E_*\quad \text{in} \ \Omega _i; \quad \Pi _hE_*\,:\!=\,E_* \quad \text{elsewhere} . \end{align*}

where $\Pi _{hi}\,:\, H^1(\Omega _i)\rightarrow V_{\mathcal{T}^i}$ is the standard Scott–Zhang interpolation operator satisfying $\Pi _{hi} E_*=E_*$ on each $\partial \Omega _i$ (see, e.g. [Reference Scott and Zhang44]). Indeed, since $E_*$ has conforming boundary values on $\partial \Omega _i$ in $V_{\mathcal{T}^i}$ , we have $\Pi _hE_*\in V_{\mathcal{T}}$ .

We denote $\psi _1=E_*-\Pi _hE_*$ . By the orthogonality and error estimate of the elliptic projection, (3.8) with $\phi =\psi _1-P_h\psi _1$ and $v_h=-P_h\psi _1\in V_{\mathcal{T}}$ , (3.2) and upper bounds (3.4)–(3.5), we derive

\begin{align*} b(E_*,E_*)&=b(E_*,E_*-\Pi _hE_*)=b(u-\widetilde u_{\mathcal{T}},\psi _1)\\ &=b(u-\widetilde u_{\mathcal{T}},\psi _1-P_h\psi _1)=b(u-u_h,\psi _1-P_h\psi _1)\\ &=a_u(u-u_h,\psi _1-P_h\psi _1)+\big ((k^2+1+k^2\varepsilon {\mathbf{1}}_{\Omega _{0}}\vert u \vert ^2)(u-u_h),\psi _1-P_h\psi _1\big )\\ &\quad -\mathrm ik\left \langle {u-u_h},{\psi _1-P_h\psi _1}\right \rangle _{\Gamma _{\textrm {imp}}} \\ &\lesssim \eta _{\mathcal R}(u_h)\left \Vert \nabla \psi _1\right \Vert _{0} +|\gamma |\eta _{\mathcal{T}}\left \Vert \nabla \psi _1\right \Vert _{0} +k^2\varepsilon \big\vert \big ((\vert u \vert ^2-\left \vert u_h \right \vert ^2\!)u_h,\psi _1-P_h\psi _1\big )_{\Omega _{0}} \big\vert \\ &\quad +k^2\left \Vert u-u_h\right \Vert _{0}h_{\mathcal{T}}^{\alpha }\left \Vert \psi _1\right \Vert _{1}+kh^{\frac {\alpha }2}_{\mathcal{T}}\left \Vert u-u_h\right \Vert _{0,\Gamma _{\textrm {imp}}}\left \Vert \psi _1\right \Vert _{1}\\ &\lesssim \eta _{\mathcal R}(u_h)\left \Vert E_*\right \Vert _{1,\Omega _{\mathcal R}}+(t(k,h_{\mathcal{T}})+|\gamma |)\eta _{\mathcal{T}}\left \Vert E_*\right \Vert _{1,\Omega _{\mathcal R}}\!. \end{align*}

This completes the proof of the lemma.

Proof. Since $u$ has a decomposition as Theorem2.4 said, the proof is similar with [Reference Duan and Wu21, Lemma 5.4] by using [Reference Gaspoz and Morin24, Theorems 5.2 and 5.3] except the estimate for $\widetilde {\textrm {osc}}_{\mathcal{T}}(V)$ : for $V\in V_{\mathcal{T}}$ ,

\begin{align*} \widetilde {\textrm {osc}}_{\mathcal{T}}^2(V)&\leq \sum _{T\in \mathcal{T}}\Big (h_T^2\big \|f+\big (k^2(1+\varepsilon {\mathbf{1}}_{\Omega _{0}}\left \vert u \right \vert ^2)+1\big )u\big \|_{0,T}^2 +h_T\big \|\mathbf{i} ku-g-\overline {\overline {\mathbf{i} ku-g}}\big \|_{0,{\partial T}\cap \Gamma _{\textrm {imp}}}^2\Big )\\ &\lesssim h^2_{\mathcal{T}}\big(\!\left \Vert f\right \Vert _{0}^2+\left \Vert g\right \Vert _{\frac 12,\Gamma _{\textrm {imp}}}^2\!\big)+\big(k^4+k^2\theta _0^2\big)h_{\mathcal{T}}^2\left \Vert u\right \Vert _{0}^2+k^2h_{\mathcal{T}}^2\left \Vert u\right \Vert _{1}^2\\ &\lesssim (k\widehat {M}(f,g) h_{\mathcal{T}})^2, \end{align*}

where the estimate is the same as in (4.8) and using (2.12). We omit the other details of the proof here.

Proof. It follows from the lower bound (3.18) and the orthogonality of the elliptic projection that

\begin{align*} \frac {1-\mu }{C_{\textrm {lb}}}\eta _{\mathcal{T}}^2&\leq (1-\mu )\big (\!\left \Vert u-\widetilde u_{\mathcal{T}}\right \Vert _{1}^2+\widetilde {\textrm {osc}}_{\mathcal{T}}^2(\widetilde u_{\mathcal{T}})\big )\\ &\leq \left \Vert u-\widetilde u_{\mathcal{T}}\right \Vert _{1}^2+\widetilde {\textrm {osc}}_{\mathcal{T}}^2(\widetilde u_{\mathcal{T}})-\big (\!\left \Vert u-\widetilde u_{\mathcal{T}_*}\right \Vert _{1}^2+\widetilde {\textrm {osc}}_{\mathcal{T}_*}^2(\widetilde u_{\mathcal{T}_*})\big )\\ &=\left \Vert \widetilde u_{\mathcal{T}_*}-\widetilde u_{\mathcal{T}}\right \Vert _{1}^2+\widetilde {\textrm {osc}}_{\mathcal{T}}^2(\widetilde u_{\mathcal{T}})-\widetilde {\textrm {osc}}_{\mathcal{T}_*}^2(\widetilde u_{\mathcal{T}_*}). \end{align*}

Since $\widetilde {\textrm {osc}}_{\mathcal{T}}(v_{\mathcal{T}})$ is independent of $v_{\mathcal{T}}$ , we have

\begin{align*} \widetilde {\textrm {osc}}_{\mathcal{T}}^2(\widetilde u_{\mathcal{T}})-\widetilde {\textrm {osc}}_{\mathcal{T}_*}^2(\widetilde u_{\mathcal{T}_*})\leq \widetilde {\textrm {osc}}_{\mathcal R}^2(\widetilde u_{\mathcal{T}}). \end{align*}

Then with the aid of (3.27), Lemma 4.6, we get

\begin{align*} \frac {1-\mu }{C_{\textrm {lb}}}\eta _{\mathcal{T}}^2&\leq C_{\textrm {lub}}\big (\eta _{\mathcal R}^2(u_h)+(t^2(k,h_{\mathcal{T}}) +|\gamma |^2)\eta ^2_{\mathcal{T}}(u_h)\big )+2\textrm {osc}_{\mathcal R}^2(u_h)+2C_{\textrm {osc}}(k^4h_{\mathcal{T}}^{2+2\alpha }+k^2h_{\mathcal{T}}^{1+\alpha })\eta _{\mathcal{T}}^2(u_h)\\ &\leq 2(C_{\textrm {lub}}+1+C_{\textrm {osc}})\big (\eta _{\mathcal R}^2(u_h)+(t^2(k,h_{\mathcal{T}}) +|\gamma |^2 +k^4h_{\mathcal{T}}^{2+2\alpha }+k^2h_{\mathcal{T}}^{1+\alpha })\eta _{\mathcal{T}}^2(u_h)\big ), \end{align*}

implying

\begin{align*} \eta _{\mathcal R}^2(u_h)&\geq \Big (\frac {1-\mu }{2C_{\textrm {lb}}(C_{\textrm {lub}}+1+C_{\textrm {osc}})}-\theta _*^2\Big )\eta _{\mathcal{T}}^2 \geq \big (2(1-\mu )\theta ^2_*-\theta _*^2\big )\eta _{\mathcal{T}}^2= \theta _{\textrm {D}}^2\eta ^2_{\mathcal{T}}(u_h), \end{align*}

where we use $t^2(k,h_{\mathcal{T}}) +|\gamma |^2 +k^4h_{\mathcal{T}}^{2+2\alpha }+k^2h_{\mathcal{T}}^{1+\alpha }\leq \theta _*^2$ since $|\gamma |$ and $k^3h_{\mathcal{T}}^{1+\alpha }$ are sufficiently small.

Proof. Denote $\epsilon ^2\,:\!=\,\mu (\!\left \Vert u-\widetilde u_n\right \Vert _{1}^2+\widetilde {\textrm {osc}}_{\mathcal{T}_n}^2), N\,:\!=\,\lceil \vert (u,f,g,k)\vert _{\frac 12}^{2}\epsilon ^{-2} \rceil$ with $\mu =\frac {\theta _*^2-\theta _{\textrm {D}}^2}{2\theta _*^2}$ , then from Lemma 4.8, there exists $\mathcal{T}_{\epsilon }\in \mathbb T, V_{\epsilon }\in V_{\mathcal{T}_{\epsilon }}$ such that

\begin{align*} \#\mathcal{T}_{\epsilon }-\#\mathcal{T}_0\leq N \quad \mbox{and}\quad N^{\frac 12}\big (\!\left \Vert u-V_{\epsilon }\right \Vert _{1}^2+\widetilde {\textrm {osc}}^2_{\mathcal{T}_{\epsilon }}(V_{\epsilon })\big )^{\frac 12}\leq \vert (u,f,g,k)\vert _{\frac 12}. \end{align*}

Let $\mathcal{T}_*\,:\!=\,\mathcal{T}_{\epsilon }\oplus \mathcal{T}_n$ (see [Reference Cascon, Kreuzer, Nochetto and Siebert13, Section 2.2]), $\widetilde u_{\mathcal{T}_*}\in V_{\mathcal{T}_*}$ is the elliptic projection of $u$ into $V_{\mathcal{T}_*}$ . It follows from Lemma 4.5 that

\begin{align*} \left \Vert u-\widetilde u_{\mathcal{T}_*}\right \Vert _{1}^2+\widetilde {\textrm {osc}}^2_{\mathcal{T}_*}(\widetilde u_{\mathcal{T}_*})&\leq \left \Vert u-V_{\epsilon }\right \Vert _{1}^2+\widetilde {\textrm {osc}}^2_{\mathcal{T}_{\epsilon }}(V_{\epsilon })\leq \mu \big (\!\left \Vert u-\widetilde u_n\right \Vert _{1}^2+\widetilde {\textrm {osc}}^2_{\mathcal{T}_n}\big ). \end{align*}

So we can use Lemma 4.9 to derive $ \eta _{\mathcal R}(u_n)\geq \theta _{\textrm {D}}\eta _{\mathcal{T}_n}(u_n)$ since $|\gamma |\leq \gamma _0$ , where $\mathcal R$ is the set of refined elements in $\mathcal{T}_n$ .

Finally, since step 3 marks $\mathcal{M}_n\subset \mathcal{T}_n$ with minimal cardinality, combining with the property of the overlay of meshes (see [Reference Cascon, Kreuzer, Nochetto and Siebert13, Lemma 3.7]) $\#\mathcal{T}_*\leq \#\mathcal{T}_\epsilon +\#\mathcal{T}_n-\#\mathcal{T}_0$ , we conclude

\begin{align*} \#\mathcal{M}_n\leq \#\mathcal R\leq \#\mathcal{T}_*-\#\mathcal{T}_n\leq \mathcal{T}_\epsilon -\#\mathcal{T}_0\leq N. \end{align*}

The proof is completed together with the definitions of $N, \epsilon$ .

Proof. Let $M=(1-\frac {\theta _{\textrm {D}}^2}{\theta _*^2})^{-1}\vert (u,f,g,k)\vert _{\frac 12}^{2}$ . Using Lemmas 4.7 and 4.10, the lower bound (3.25), the equivalence of $\widetilde \eta _{\mathcal{T}}$ and $\widetilde \eta _{\mathcal{T}}^*$ and the contraction property (4.7), we derive

\begin{align*} \#\mathcal{T}_n-\#\mathcal{T}_0&\lesssim \sum _{j=0}^{n-1}\#\mathcal{M}_j\lesssim M\sum _{j=0}^{n-1}\big (\!\left \Vert u-\widetilde u_j\right \Vert _{1}^2+\widetilde {\textrm {osc}}^2_{\mathcal{T}_j}\big )^{-1}\\ &\lesssim M\sum _{j=0}^{n-1}\big (\!\left \Vert u-\widetilde u_j\right \Vert _{1}^2+\beta \widetilde \eta _{\mathcal{T}_j}^2\big )^{-1}\eqsim M\sum _{j=0}^{n-1}\big (\!\left \Vert u-\widetilde u_j\right \Vert _{1}^2+\beta (\widetilde \eta _{\mathcal{T}_j}^*)^2\big )^{-1}\\ &\lesssim M\sum _{j=1}^{n}C_{\textrm {conv}}^{2j}\big (\!\left \Vert u-\widetilde u_n\right \Vert _{1}^2+\beta \big(\widetilde \eta ^*_{\mathcal{T}_n}\big)^2\big )^{-1}\lesssim \frac {MC_{\textrm {conv}}^{2}}{1-C_{\textrm {conv}}^{2}}\big (\!\left \Vert u-\widetilde u_n\right \Vert _{1}^2+\beta \widetilde \eta ^2_{\mathcal{T}_n}\big )^{-1}. \end{align*}

Finally, from upper bounds (3.4) and (3.3), Corollary 3.9 and Lemma 4.8, we have

\begin{align*} \left\Vert {\hskip -4.1pt}\left\vert u-u_n \right\vert {\hskip -4.1pt}\right\Vert &\lesssim (1+k^2h_n)\frac {C_{\textrm {conv}}}{(1-C_{\textrm {conv}}^{2})^{\frac 12}}M^{\frac 12}(\#\mathcal{T}_n-\#\mathcal{T}_0)^{-\frac 12}\\ &\lesssim (1+k^2h_n)k\widehat {M}(f,g)(\#\mathcal{T}_n-\#\mathcal{T}_0)^{-\frac 12}. \end{align*}

Remark 4.12. When the pollution error dominates the error, the mesh is quasi-uniformly refined numerically, so the second term in (4.10) is nearly $k^3(\#\mathcal{T}_n-\#\mathcal{T}_0)^{-1}$ , which is the quasi-optimal decay rate of the pollution error. Once the pollution error disappears, the first term dominates the upper bound in (4.10), which seems as the interpolation error of the regular part and it is quasi-optimal.