Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations II: HHO-Inspired Methods

Abstract

In Chen et al. (J. Sci. Comput. 81(3): 2188–2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction-diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree $k\geqslant 1$. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3): 635–650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements affine-equivalent to a finite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for $k\geqslant 0$ by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

Appendices

Appendix A Approximation Estimates of Auxiliary Projections

1.1 A.1 Proof of (10a)

Here we prove the estimate for $\varPi _{k+1}^\star u- u$ in (10a). We are going to use the following auxiliary result.

Lemma A1

For any $K\in \mathcal {T}_h$, we have

$$\begin{aligned} \Vert \varPi _{k+1}^\star u-u\Vert _{K} \leqslant C\left( h\Vert \nabla u-\nabla \varPi ^{\text{o}}_{k+1}u\Vert _{K}+\Vert u-\varPi ^{{\text{o}}}_{k+1}u\Vert _{K}\right) . \end{aligned}$$

Proof

By definitions (3) and (4), we obtain

$$\begin{aligned} (\nabla \varPi _{k+1}^\star u,\nabla z_h)_K&=-(\varPi ^{{\text{o}}}_\ell u,\varDelta z_h)_K+\langle \varPi ^{\partial }_k u,\varvec{n}\cdot \nabla z_h \rangle _{\partial K},\\ (\varPi _{k+1}^\star u,w_h)_K&=(\varPi ^{{\text{o}}}_\ell u,w_h)_K \end{aligned}$$

for all $(z_h,w_h)\in [\mathcal {P}_{\ell }^{k+1}(K)]^\perp \times \mathcal {P}^{\ell }(K)$. This leads to

$$\begin{aligned} (\nabla \varPi _{k+1}^\star u,\nabla z_h)_K&=(\nabla u,\nabla z_h)_K,\\ (\varPi _{k+1}^\star u,w_h)_K&=(\varPi ^{{\text{o}}}_{k+1} u,w_h)_K. \end{aligned}$$

The last equation implies that $\varPi _{k+1}^\star u-\varPi ^{{\text{o}}}_{k+1}u\in [\mathcal {P}_{\ell }^{k+1}(K)]^{\perp }$ and so, we can then take $z_h:=\varPi _{k+1}^\star u-\varPi ^{{\text{o}}}_{k+1}u$ in the first equation to get

$$\begin{aligned} \Vert \nabla \varPi _{k+1}^\star u-\nabla \varPi ^{{\text{o}}}_{k+1}u\Vert ^2_{K}=(\nabla \varPi _{k+1}^\star u-\nabla \varPi ^{{\text{o}}}_{k+1}u, \nabla u-\nabla \varPi ^{{\text{o}}}_{k+1}u)_K, \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla \varPi _{k+1}^\star u-\nabla \varPi ^{{\text{o}}}_{k+1}u\Vert _{K}\leqslant \Vert \nabla u-\nabla \varPi ^{{\text{o}}}_{k+1}u\Vert _{K}. \end{aligned}$$

Since $\varPi _{k+1}^\star u-\varPi ^{{\text{o}}}_{k+1}u\in [\mathcal {P}_{\ell }^{k+1}(K)]^{\perp }$, we have

$$\begin{aligned} (\varPi _{k+1}^\star u-\varPi ^{{\text{o}}}_{k+1}u,1)_K=0, \end{aligned}$$

and using Poincaré’s inequality, we obtain

$$\begin{aligned} \Vert \varPi _{k+1}^\star u-\varPi ^{{\text{o}}}_{k+1}u\Vert _{K} \leqslant C h\Vert \nabla \varPi _{k+1}^\star u-\nabla \varPi ^{{\text{o}}}_{k+1}u\Vert _{K} \leqslant C h\Vert \nabla u-\nabla \varPi ^{{\text{o}}}_{k+1}u\Vert _{K}. \end{aligned}$$

Then the estimate follows by applying the triangle inequality. This completes the proof.

We are now ready to prove (10a). Using inverse inequalities, Poincaré’s inequality, and the approximation properties for $\varPi _{k+1}^{\text{o}}$, one gets

$$\begin{aligned} \begin{aligned} \Vert u-\varPi _{k+1}^{\star }u\Vert _{0,\infty ,K}&\leqslant \Vert \varPi _{k+1}^{\star }u-\varPi _{k+1}^{\text{o}} u\Vert _{0,\infty ,K} + \Vert \varPi _{k+1}^{\text{o}} u-u\Vert _{0,\infty ,K}\\&\leqslant Ch^{-d/2}\Vert \varPi _{k+1}^{\star }u-\varPi _{k+1}^{\text{o}} u\Vert _{0,K}+ \Vert \varPi _{k+1}^{\text{o}} u-u\Vert _{0,\infty ,K}\\&\leqslant Ch^{1-d/2}| u-\varPi _{k+1}^{\text{o}} u|_{1,K}+ \Vert \varPi _{k+1}^{\text{o}} u-u\Vert _{0,\infty ,K}. \end{aligned} \end{aligned}$$

(A1)

By [1, Lemma 4.3.8], there exists $Q^{k+1} u\in \mathcal {P}^{k+1}(K)$ such that

$$\begin{aligned} \Vert Q^{k+1} u - u\Vert _{1, \infty ,K}\leqslant C |u|_{1,\infty ,K},\quad \Vert Q^{k+1} u - u\Vert _{0, \infty ,K}\leqslant C h|u|_{1,\infty ,K}. \end{aligned}$$

Hence, by (A1) we have

$$\begin{aligned}&\Vert u-\varPi _{k+1}^{\star }u\Vert _{0,\infty ,K}\\& \leqslant Ch^{1-d/2} \left( \Vert u - Q^{k+1} u \Vert _{1,K} + \Vert Q^{k+1} u-\varPi _{k+1}^{\text{o}} u\Vert _{1,K}\right) \\& +\left( \Vert u - Q^{k+1} u \Vert _{0,\infty ,K} + \Vert Q^{k+1} u-\varPi _{k+1}^{\text{o}} u\Vert _{0,\infty ,K}\right) \\&= Ch^{1-d/2} \left( \Vert u - Q^{k+1} u \Vert _{1,K} + \Vert \varPi _{k+1}^{\text{o}} (Q^{k+1} u- u)\Vert _{1,K}\right) \\& +C\left( \Vert u - Q^{k+1} u \Vert _{0,\infty ,K} + h^{-d/2}\Vert Q^{k+1} u-\varPi _{k+1}^{\text{o}} u\Vert _{K}\right) \\&\leqslant Ch^{1-d/2} \left( \Vert u - Q^{k+1} u \Vert _{1,K} + \Vert \varPi _{k+1}^{\text{o}} (Q^{k+1} u- u)\Vert _{1,K}\right) \\& +C\left( \Vert u - Q^{k+1} u \Vert _{0,\infty ,K} + h^{-d/2}\Vert \varPi _{k+1}^{\text{o}}(Q^{k+1} u- u)\Vert _{K}\right) \\&\leqslant Ch^{1-d/2} \left( \Vert u - Q^{k+1} u \Vert _{1,K} + \Vert Q^{k+1} u- u\Vert _{1,K}\right) \\& +C\left( \Vert u - Q^{k+1} u \Vert _{0,\infty ,K} + h^{-d/2}\Vert Q^{k+1} u- u\Vert _{K}\right) \\&\leqslant Ch^{1-d/2}\Vert u - Q^{k+1} u \Vert _{1,K} + Ch^{-d/2}\Vert u - Q^{k+1} u \Vert _{K} +C\Vert u - Q^{k+1} u \Vert _{0,\infty ,K}\\&\leqslant C h\Vert u - Q^{k+1} u \Vert _{1,\infty ,K} +C \Vert u - Q^{k+1} u \Vert _{0,\infty ,K}\\&\leqslant C h |u|_{1,\infty ,K}. \end{aligned}$$

This completes the proof of (10a).

1.2 A.2 Proof of (10b)

Here, we prove the estimate for $\varPi _{k+1}^\star u - u_h^\star$ in (10b).

Let $z_h\in [\mathcal {P}_{\ell }^{k+1}(K)]^{\perp }$ and take $\varvec{r}_h=\nabla z_h$ in the first equation of Proposition 1 to get

$$\begin{aligned} (\varvec{q}_h,\nabla z_h)-(u_h,\Delta z_h)_{\mathcal {T}_h}+\left\langle \widehat{u}_h,\nabla z_h\cdot \varvec{n} \right\rangle _{\partial {\mathcal {T}_h}} = 0. \end{aligned}$$

Combined with (4a) one gets

$$\begin{aligned} (\nabla u_h^{\star },\nabla z_h)=-(\varvec{q}_h,\nabla z_h), \quad \forall z_h\in [\mathcal {P}_{\ell }^{k+1}(K)]^{\perp }. \end{aligned}$$

By the definition of $\varPi _{k+1}^{\star }$, as in the proof of Proposition 1 one gets

$$\begin{aligned} ( \nabla \varPi _{k+1}^{\star }u ,\nabla z_{h} )_K&=-(\varPi ^{{\text{o}}}_\ell u,\varDelta z_h)_K+\langle \varPi ^{\partial }_k u,\varvec{n}\cdot \nabla z_h \rangle _{\partial K}=(\nabla u,\nabla z_h)_K. \end{aligned}$$

Let $e_h=u_h^\star - u_h+\varPi _{\ell }^{\text{o}} u-\varPi _{k+1}^{\star } u$. Then $e_h\in [\mathcal {P}_{\ell }^{k+1}(K)]^{\perp }$. By the two previous equations, $\varvec{q} = -\nabla u$, and an inverse inequality we have

$$\begin{aligned} \Vert \nabla e_{h}\Vert _K^2&=(\nabla (u_h^\star - u_h),\nabla e_{h} )_K+( \nabla (\varPi _{\ell }^{\text{o}} u-\varPi _{k+1}^{\star } u),\nabla e_{h} )_K\\&=(-\varvec{q}_h-\nabla u_h,\nabla e_{h} )_K+( \nabla (\varPi _{\ell }^{\text{o}} u-u),\nabla e_{h} )_K \\&=((\varvec{q}-\varvec{\varPi }^{\text{o}}_k\varvec{q})-(\varvec{q}_h-\varvec{\varPi }^{\text{o}}_k\varvec{q}) +\nabla (\varPi _{\ell }^{\text{o}} u-u_h),\nabla e_{h})_K \\&\leqslant C (h^{-1}\Vert u_h-\varPi _{\ell }^{\text{o}} u\Vert _K+\Vert \varvec{q}_h-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _K + \Vert \varvec{q}-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _K )\Vert \nabla e_{h}\Vert _K. \end{aligned}$$

Since $(e_h,1)_K=0$, we can now apply the Poincaré inequality to get

$$\begin{aligned} \Vert e_{h}\Vert _K \leqslant C h \Vert \nabla e_h\Vert _K \leqslant C (\Vert u_h-\varPi _{\ell }^{\text{o}} u\Vert _K+h\Vert \varvec{q}_h-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _K + h\Vert \varvec{q}-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _K). \end{aligned}$$

This means

$$\begin{aligned} \Vert e_{h} \Vert _{\mathcal {T}_h} \leqslant C( \Vert u_h-\varPi _{\ell }^{\text{o}} u\Vert _{\mathcal {T}_h}+h\Vert \varvec{q}_h-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _{\mathcal {T}_h} +h\Vert \varvec{q}-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _{\mathcal {T}_h} ). \end{aligned}$$

Hence, we have

$$\begin{aligned} \Vert \varPi _{k+1}^{\star } u - u_h^\star \Vert _{\mathcal {T}_h}&\leqslant \Vert \varPi _{k+1}^{\star } u -\varPi ^{{\text{o}}}_{\ell } u- u_h^\star + u_h\Vert _{\mathcal {T}_h} + \Vert \varPi ^{{\text{o}}}_{\ell } u-u_h\Vert _{\mathcal {T}_h} \\&\leqslant C ( \Vert u_h-\varPi _{\ell }^{\text{o}} u\Vert _{\mathcal {T}_h}+h\Vert \varvec{q}_h-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _{\mathcal {T}_h} +h\Vert \varvec{q}-\varvec{\varPi }^{\text{o}}_k\varvec{q}\Vert _{\mathcal {T}_h} ). \end{aligned}$$

This completes the proof of (10b).

Appendix B Proof of Theorem 2

This appendix is devoted to the proof of the approximation estimates of Theorem 2. We only give the proofs of the estimates for $\Vert \varvec{\varPi }_{k}^{\text{o}} \varvec{q} - \overline{\varvec{q}}_h\Vert _{\mathcal {T}_h}$ and $\Vert \varPi _{\ell }^{\text{o}}u-\overline{u}_h\Vert _{\mathcal {T}_h}$. The proof of the estimate for $\Vert \partial _t\varPi _{\ell }^{\text{o}}u-\partial _t\overline{u}_h\Vert _{\mathcal {T}_h}$ is very similar and is omitted. We use the notation

$$\begin{aligned} \varepsilon _h^{\varvec{q}}=\varvec{\varPi }_{k}^{\text{o}}\varvec{q}-\overline{\varvec{q}}_h , \quad \varepsilon _h^{ u}=\varPi _{\ell }^{\text{o}} u-\overline{u}_h, \quad \varepsilon _h^{\widehat{u}}=\varPi _{k}^{\partial }u-\widehat{\overline{u}}_h, \quad \text { and } \quad \varepsilon _h^{ u^\star }=\varPi _{k+1}^\star u-\overline{u}_h^\star , \end{aligned}$$

and split the proof into four steps.

1.1 Step 1: Equations for the projections of the errors

Lemma B1

For all $(\varvec{r}_h,v_h,\widehat{v}_h)\in \varvec{V}_h\times W_h\times M_h$, we have

$$\begin{aligned}&(\varepsilon _h^{\varvec{q}},\varvec{r}_h)_{\mathcal {T}_h}-(\varepsilon _h^{u},\nabla \cdot \varvec{r}_h)_{\mathcal {T}_h}+\langle \varepsilon _h^{\widehat{u}},\varvec{r}_h\cdot \varvec{n}\rangle _{\partial {\mathcal {T}_h}} = 0,\\&\quad (\nabla \cdot \varepsilon _h^{\varvec{q}}, v_h)_{\mathcal {T}_h} -\langle \varepsilon _h^{\varvec{q}}\cdot \varvec{n},\widehat{v}_h \rangle _{\partial {\mathcal {T}_h}} +\langle h^{-1}( \varPi ^{\partial }_k \varepsilon _h^{u^\star } -\varepsilon _h^{\widehat{u}}),\varPi ^{\partial }_kv_h^\star -\widehat{v}_h\rangle _{\partial {\mathcal {T}_h}} = RHS_h, \end{aligned}$$

where

$$\begin{aligned} RHS_h:=&\;((\mathbb {I}-\varPi _{\ell }^{\text{o}})(-\Delta u),(\mathbb {I}-\varPi _{\ell }^{\text{o}})v_h^\star ) +E_h(\varvec{q},u;v_h,\widehat{v}_h),\\ E_h(\varvec{q},u;v_h,\widehat{v}_h) :=&-\langle ( \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q})\cdot \varvec{n},\widehat{v}_h-v_h^\star \rangle _{\partial {\mathcal {T}_h}} +\langle h^{-1}( \varPi _{k+1}^\star u -u),\varPi ^{\partial }_k v_h^\star -\widehat{v}_h\rangle _{\partial {\mathcal {T}_h}}, \end{aligned}$$

and $\mathbb {I}$ is the identity operator.

Proof

We begin by noting that, by the properties of $\varvec{\varPi }_k^{\text{o}}$, $\varPi _{\ell }^{\text{o}}$, and $\varPi _k^\partial$, we have

$$\begin{aligned}&(\varvec{\varPi }^{\text{o}}_k\varvec{q},\varvec{r}_h)_{\mathcal {T}_h}-(\varPi ^{\text{o}}_{\ell } u,\nabla \cdot \varvec{r}_h)_{\mathcal {T}_h}+\langle \varPi ^{\partial }_{k}u,\varvec{r}_h\cdot \varvec{n} \rangle _{\partial {\mathcal {T}_h}}\\&\quad =(\varvec{q},\varvec{r}_h)_{\mathcal {T}_h}-( u,\nabla \cdot \varvec{r}_h)_{\mathcal {T}_h}+\left\langle u,\varvec{r}_h\cdot \varvec{n} \right\rangle _{\partial {\mathcal {T}_h}}=0, \end{aligned}$$

since $\varvec{q}+\nabla u=0$. Also, since $\langle \varvec{q}\cdot \varvec{n},\widehat{v}_h \rangle _{\partial {\mathcal {T}_h}}=0$, we have

$$\begin{aligned} (\nabla \cdot \varvec{\varPi }^{\text{o}}_k\varvec{q}, v_h)_{\mathcal {T}_h} -\langle \varvec{\varPi }^{\text{o}}_k\varvec{q}\cdot \varvec{n},\widehat{v}_h \rangle _{\partial {\mathcal {T}_h}}&=(\nabla \cdot \varvec{\varPi }^{\text{o}}_k\varvec{q}, v_h^\star )_{\mathcal {T}_h} -\langle \varvec{\varPi }^{\text{o}}_k\varvec{q}\cdot \varvec{n},\widehat{v}_h \rangle _{\partial {\mathcal {T}_h}}\\&= (\nabla \cdot \varvec{q}, v^\star _h)_{\mathcal {T}_h} -\langle ( \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q})\cdot \varvec{n},\widehat{v}_h -v_h^\star \rangle _{\partial {\mathcal {T}_h}}\\&=(-\Delta u, v^\star _h)_{\mathcal {T}_h} -\langle ( \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q})\cdot \varvec{n},\widehat{v}_h -v_h^\star \rangle _{\partial {\mathcal {T}_h}}. \end{aligned}$$

As a consequence,

$$\begin{aligned}&(\varvec{\varPi }^{\text{o}}_k\varvec{q},\varvec{r}_h)_{\mathcal {T}_h}-(\varPi ^{\text{o}}_{\ell } u,\nabla \cdot \varvec{r}_h)_{\mathcal {T}_h}+\langle \varPi ^{\partial }_{k}u,\varvec{r}_h\cdot \varvec{n} \rangle _{\partial {\mathcal {T}_h}} = 0,\\& (\nabla \cdot \varvec{\varPi }^{\text{o}}_k\varvec{q}, v_h)_{\mathcal {T}_h}-\langle \varvec{\varPi }^{\text{o}}_k\varvec{q}\cdot \varvec{n},\widehat{v}_h \rangle _{\partial {\mathcal {T}_h}} +\langle h^{-1}( \varPi ^{\partial }_k \varPi _{k+1}^\star u -\varPi ^{\partial }_{k}u),\varPi ^{\partial }_kv_h^\star -\widehat{v}_h\rangle _{\partial {\mathcal {T}_h}}\\& =(-\Delta u,v_h^\star )_{\mathcal {T}_h} +E_h(\varvec{q},u;v_h,\widehat{v}_h). \end{aligned}$$

The wanted equations can be now obtained by subtracting these equations from the equations defining the HDG elliptic approximation (7). This completes the proof.

1.2 Step 2: Estimate for $\varepsilon _h^q$ by an energy argument

Lemma B2

We have

$$\begin{aligned}&\Vert \nabla \varepsilon _h^{u^\star }\Vert _{\mathcal {T}_h} +\Vert \varepsilon _h^{\varvec{q}}\Vert _{\mathcal {T}_h} +\Vert h^{-1/2}(\varPi ^{\partial }_k\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}})\Vert _{\partial \mathcal {T}_h}\\& \leqslant C\left( h\Vert (\varPi _{\ell }^{{\text{o}}}-\mathbb {I})(-\Delta u)\Vert _{\mathcal {T}_h} +h^{1/2}\Vert \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q}\Vert _{\partial \mathcal {T}_h}+ \Vert h^{-1/2}(\varPi _{k+1}^\star u -u)\Vert _{\partial {\mathcal {T}_h}} \right) . \end{aligned}$$

This result implies the estimate for the approximate flux in Theorem 2. To prove this lemma, we need the following auxiliary result.

Lemma B3

We have

$$\begin{aligned} \Vert \varepsilon _h^{\varvec{q}}\Vert _{\mathcal {T}_h}&\leqslant C\left( \Vert \nabla \varepsilon _h^{u^\star }\Vert _{\mathcal {T}_h} +\Vert h^{-1/2}(\varPi _{k}^{\partial }\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}})\Vert _{\partial \mathcal {T}_h}\right) , \end{aligned}$$

(B1a)

$$\begin{aligned} \Vert \nabla \varepsilon _h^{u^\star }\Vert _{\mathcal {T}_h}&\leqslant \Vert \varepsilon _h^{\varvec{q}}\Vert _{\mathcal {T}_h} +\Vert h^{-1/2}(\varPi _{k}^{\partial }\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}})\Vert _{\partial \mathcal {T}_h} . \end{aligned}$$

(B1b)

Proof

Using the first equation of Lemma B1, the definition of $\mathfrak {p}_h^{k+1}$ in (4), and $\nabla \cdot \varvec{r}_h\in W_h$, we have

$$\begin{aligned} (\varepsilon _h^{\varvec{q}},\varvec{r}_h)_{\mathcal {T}_h}-(\varepsilon _h^{u^\star },\nabla \cdot \varvec{r}_h)_{\mathcal {T}_h}+\langle \varepsilon _h^{\widehat{u}},\varvec{r}_h\cdot \varvec{n} \rangle _{\partial {\mathcal {T}_h}} = 0. \end{aligned}$$

Integration by parts gives

$$\begin{aligned} (\varepsilon _h^{\varvec{q}},\varvec{r}_h)_{\mathcal {T}_h}+(\nabla \varepsilon _h^{u^\star }, \varvec{r}_h)_{\mathcal {T}_h}+\langle \varepsilon _h^{\widehat{u}}-\varPi _{k}^{\partial }\varepsilon _h^{u^\star },\varvec{r}_h\cdot \varvec{n} \rangle _{\partial {\mathcal {T}_h}} = 0. \end{aligned}$$

Since $\nabla \varepsilon _h^{u^*}\in \varvec{V}_h$, by taking first $\varvec{r}_h:=\varepsilon _h^{\varvec{q}}$ and then $\varvec{r}_h:=\nabla \varepsilon _h^{u^*}$, one gets

$$\begin{aligned} \Vert \varepsilon _h^{\varvec{q}}\Vert _{\mathcal {T}_h}&\leqslant C\left( \Vert \nabla \varepsilon _h^{u^\star }\Vert _{\mathcal {T}_h} +\Vert h^{-1/2}(\varPi _{k}^{\partial }\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}})\Vert _{\partial \mathcal {T}_h}\right) ,\\ \Vert \nabla \varepsilon _h^{u^\star }\Vert _{\mathcal {T}_h}&\leqslant C\left( \Vert \varepsilon _h^{\varvec{q}}\Vert _{\mathcal {T}_h} +\Vert h^{-1/2}(\varPi _{k}^{\partial }\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}})\Vert _{\partial \mathcal {T}_h}\right) , \end{aligned}$$

respectively. This completes the proof.

We can now prove Lemma B2.

Proof

We take $(\varvec{r}_h,v_h,\widehat{v}_h):=(\varepsilon _h^{\varvec{q}},\varepsilon _h^{u},\varepsilon _h^{\widehat{u}})$ in the error equations of Lemma B1, and add them to get

$$\begin{aligned} \Vert \varepsilon _h^{\varvec{q}}\Vert ^2_{\mathcal {T}_h}+\Vert h^{-1/2}( \varPi ^{\partial }_k \varepsilon _h^{u^\star } -\varepsilon _h^{\widehat{u}})\Vert ^2_{\partial {\mathcal {T}_h}} = R_1+R_2+R_3, \end{aligned}$$

where

$$\begin{aligned} R_1&:= ((\mathbb {I}-\varPi _{\ell }^{\text{o}})(-\Delta u), (\mathbb {I}-\varPi _{\ell }^{\text{o}})\varepsilon _h^{u^\star })_{\mathcal {T}_{h}},\\ R_2&:=-\langle ( \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q})\cdot \varvec{n},\varepsilon ^{\widehat{u}}_h-\varepsilon _h^{u^\star } \rangle _{\partial {\mathcal {T}_h}}\\ R_3&:=\langle h^{-1}( \varPi _{k+1}^\star u -u),\varPi ^{\partial }_k\varepsilon _h^{u^\star } -\varepsilon ^{\widehat{u}}_h\rangle _{\partial \mathcal {T}_h}. \end{aligned}$$

Since

$$\begin{aligned} |R_1|&\leqslant Ch\Vert (\mathbb {I}-\varPi _{\ell }^{{\text{o}}})(-\Delta u)\Vert _{\mathcal {T}_h}\Vert \nabla \varepsilon _h^{u^\star } \Vert _{\mathcal {T}_h}, \\ |R_2|&\leqslant Ch^{1/2}\Vert \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q}\Vert _{\partial \mathcal {T}_h} \left( \Vert \nabla \varepsilon _h^{u^\star } \Vert _{\mathcal {T}_h}+\Vert h^{-1/2}( \varPi ^{\partial }_k \varepsilon _h^{u^\star } -\varepsilon _h^{\widehat{u}})\Vert _{\partial {\mathcal {T}_h}} \right) ,\\ |R_3|&\leqslant \Vert h^{-1/2}(\varPi _{k+1}^\star u -u)\Vert _{\partial {\mathcal {T}_h}}\Vert h^{-1/2}( \varPi ^{\partial }_k \varepsilon _h^{u^\star } -\varepsilon _h^{\widehat{u}})\Vert _{\partial {\mathcal {T}_h}}, \end{aligned}$$

using the last two estimates of Lemma B3 and simple algebraic manipulations, we get the desired result.

1.3 Step 3: Estimate for $\varepsilon _h^{u^\star }$ by a duality argument

Lemma B4

Assume that the elliptic regularity inequality (9a) holds. Then, we have

$$\begin{aligned} {\Vert \varepsilon _h^{u^\star }\Vert _{\mathcal {T}_h}}&\leqslant C h^{1+\min \{\ell ,1\}}\Vert (\mathbb {I}- \varPi _{\ell }^{\text{o}})(-\Delta u)\Vert _{\mathcal {T}_h}\\&\quad +C(h^{3/2}\Vert \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q}\Vert _{\partial \mathcal {T}_h}+ h\Vert h^{-1/2}(\varPi _{k+1}^\star u -u)\Vert _{\partial {\mathcal {T}_h}}). \end{aligned}$$

Proof

Setting $g:=\varepsilon _h^{u^\star }$ in the dual problem, and proceeding as in the proof of Lemma B1, we get

$$\begin{aligned}&(\varvec{\varPi }^{\text{o}}_k\varvec{\varPhi },\varvec{r}_h)_{\mathcal {T}_h}-(\varPi ^{\text{o}}_{\ell } \varPsi ,\nabla \cdot \varvec{r}_h)_{\mathcal {T}_h}+\langle \varPi ^{\partial }_{k}\varPsi ,\varvec{r}_h\cdot \varvec{n} \rangle _{\partial {\mathcal {T}_h}} = 0, \end{aligned}$$

(B2a)

$$\begin{aligned}&\quad (\nabla \cdot \varvec{\varPi }^{\text{o}}_k\varvec{\varPhi }, v_h)_{\mathcal {T}_h} -\langle \varvec{\varPi }^{\text{o}}_k\varvec{\varPhi }\cdot \varvec{n},\widehat{v}_h \rangle _{\partial {\mathcal {T}_h}} \nonumber \\&\quad +\langle h^{-1}( \varPi ^{\partial }_k \varPi _{k+1}^\star \varPsi -\varPi ^{\partial }_{k}\varPsi ),\varPi ^{\partial }_kv_h^\star -\widehat{v}_h\rangle _{\partial {\mathcal {T}_h}} =(\varepsilon _h^{u^\star },v^\star _h)_{\mathcal {T}_h}+E_h(\varvec{\varPhi },\varPsi ;v_h,\widehat{v}_h), \end{aligned}$$

(B2b)

where

$$\begin{aligned} E_h(\varvec{\varPhi },\varPsi ;v_h,\widehat{v}_h)=-\langle ( \varvec{\varPi }^{\text{o}}_k\varvec{\varPhi }-\varvec{\varPhi })\cdot \varvec{n},\widehat{v}_h-v_h^\star \rangle _{\partial {\mathcal {T}_h}}+\langle h^{-1}( \varPi _{k+1}^\star {\varPsi } -\varPsi ),\varPi ^{\partial }_kv_h^\star -\widehat{v}_h\rangle _{\partial {\mathcal {T}_h}}. \end{aligned}$$

Then taking $(v_h,\widehat{v}_h):=(\varepsilon _h^u,\varepsilon _h^{\widehat{u}})$ in (B2b), we get

$$\begin{aligned} \Vert \varepsilon _h^{u^\star }\Vert ^2_{\mathcal {T}_h}&=(\nabla \cdot \varvec{\varPi }^{\text{o}}_k\varvec{\varPhi }, \varepsilon _h^u)_{\mathcal {T}_h} -\langle \varvec{\varPi }^{\text{o}}_k\varvec{\varPhi }\cdot \varvec{n},\varepsilon _h^{\widehat{u}} \rangle _{\partial {\mathcal {T}_h}}\\&\quad +\langle h^{-1}( \varPi ^{\partial }_k \varPi _{k+1}^\star \varPsi -\varPi ^{\partial }_{k}\varPsi ),\varPi ^{\partial }_kv_h^\star -\widehat{v}_h\rangle _{\partial {\mathcal {T}_h}}{-E_h(\varvec{\varPhi },\varPsi ;\varepsilon ^{u}_h,\varepsilon ^{\widehat{u}}_h)} \\&= (\varepsilon _h^{\varvec{q}},\varvec{\varPi }^{\text{o}}_k\varvec{\varPhi })_{\mathcal {T}_h} + \langle h^{-1}( \varPi ^{\partial }_k \varPi _{k+1}^\star \varPsi -\varPi ^{\partial }_{k}\varPsi ),\varPi ^{\partial }_k\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}}\rangle _{\partial {\mathcal {T}_h}}{-E_h(\varvec{\varPhi },\varPsi ;\varepsilon ^{u}_h,\varepsilon ^{\widehat{u}}_h)} \end{aligned}$$

by the first equation of Lemma B1 with $\varvec{r}_h:=\varvec{\varPi }^{\text{o}}_k\varvec{\varPhi }$. By (B2a) with $\varvec{r}_h:=\varepsilon _h^{\varvec{q}}$, we obtain

$$\begin{aligned} \Vert \varepsilon _h^{u^\star }\Vert ^2_{\mathcal {T}_h}&= (\varPi ^{\text{o}}_{\ell } \varPsi ,\nabla \cdot \varepsilon _h^{\varvec{q}})_{\mathcal {T}_h}\ -\langle \varPi ^{\partial }_{k}\varPsi ,\varepsilon _h^{\varvec{q}}\cdot \varvec{n} \rangle _{\partial {\mathcal {T}_h}} \\&\quad +\langle h^{-1}( \varPi ^{\partial }_k \varPi _{k+1}^\star \varPsi -\varPi ^{\partial }_{k}\varPsi ),\varPi ^{\partial }_k\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}}\rangle _{\partial {\mathcal {T}_h}}\\&\quad -E_h(\varvec{\varPhi },\varPsi ; \, \varepsilon ^{u}_h,\varepsilon ^{\widehat{u}}_h)\\&=({(\mathbb {I}-\varPi _{\ell }^{\text{o}})(-\Delta u)},\varPi _{k+1}^\star \varPsi -\varPi _{\ell }^{\text{o}}\varPsi )\\&\quad + E_h(\varvec{q},u; \, \varPi ^{{\text{o}}}_{\ell }\varPsi ,\varPi ^{\partial }_{k}\varPsi )-E_h(\varvec{\varPhi },\varPsi ; \, \varepsilon ^{u}_h,\varepsilon ^{\widehat{u}}_h) \end{aligned}$$

by the second equation of Lemma B1 with $(v_h,\widehat{v}_h):=(\varPi ^{{\text{o}}}_{\ell }\varPsi ,\varPi ^{\partial }_{k}\varPsi )$. Inserting the definitions of the $E_h$-terms, we finally get

$$\begin{aligned} \Vert \varepsilon _h^{u^\star }\Vert ^2_{\mathcal {T}_h}&= ((\mathbb {I}-\varPi _{\ell }^{\text{o}})(-\Delta u),\varPi _{k+1}^\star \varPsi -\varPi _{\ell }^{\text{o}}\varPsi )\\&\quad -\langle ( \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q})\cdot \varvec{n},\varPi ^{\partial }_{k}\varPsi -\varPi ^\star _{k+1}\varPsi \rangle _{\partial {\mathcal {T}_h}} \\&\quad +\langle h^{-1}( \varPi _{k+1}^\star u -u),\varPi ^{\partial }_k\varPi ^\star _{k+1}\varPsi -\varPi ^{\partial }_{k}\varPsi \rangle _{\partial {\mathcal {T}_h}}\\&\quad +\langle ( \varvec{\varPi }^{\text{o}}_k\varvec{\varPhi }-\varvec{\varPhi })\cdot \varvec{n},\varepsilon _h^{\widehat{u}}-\varepsilon _h^{u^\star }\rangle _{\partial {\mathcal {T}_h}}\\&\quad -\langle h^{-1}( \varPi _{k+1}^\star {\varPsi } -\varPsi ),\varPi ^{\partial }_k\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}}\rangle _{\partial {\mathcal {T}_h}}, \end{aligned}$$

which leads to

$$\begin{aligned} \Vert \varepsilon _h^{u^\star }\Vert ^2_{\mathcal {T}_h}&\leqslant C h^{\min \{\ell ,1\}+1}\Vert (\mathbb {I}- \varPi _{\ell }^{\text{o}})(-\Delta u)\Vert _{\mathcal {T}_h}|\varPsi |_{\min \{\ell ,1\}+1} \\&\quad +Ch^{3/2}\Vert \varvec{\varPi }^{\text{o}}_k\varvec{q}-\varvec{q}\Vert _{\partial \mathcal {T}_h} |\varPsi |_2 +Ch\Vert h^{-1/2}(\varPi _{k+1}^\star u -u)\Vert _{\partial {\mathcal {T}_h}}{|\varPsi |_2}\\&\quad +Ch\left( \Vert \nabla \varepsilon _h^{u^\star }\Vert _{\mathcal {T}_h} +\Vert h^{-1/2}(\varPi ^{\partial }_k\varepsilon _h^{u^\star }-\varepsilon _h^{\widehat{u}})\Vert _{\partial \mathcal {T}_h} \right) ( |\varvec{\varPhi }|_1+|\varPsi |_2). \end{aligned}$$

Using the elliptic regularity inequality (9a) and the first inequality of Lemma B2, we finally obtain the wanted result.

1.4 Step 4: Estimate for $u_h$

Lemma B5

We have that $\Vert \varepsilon _h^{u} \Vert _{\mathcal {T}_h} \leqslant \Vert \varepsilon _h^{u^\star } \Vert _{\mathcal {T}_h}.$

Combining this result and the one in the previous step gives the estimate in the approximation for u in Theorem 2. To complete the proof of Theorem 2, it only remains to prove the above lemma.

Proof

Since $u_h^\star =\mathfrak {p}_h^{k+1}( u_h,\widehat{u}_h)$, $\varPi _{k+1}^\star u=\mathfrak {p}_h^{k+1}(\varPi ^{{\text{o}}}_{\ell } u,\varPi ^{\partial }_k u)$, and the operator $\mathfrak {p}_h^{k+1}$ is linear, we have that $\varepsilon _h^{u^\star } =\mathfrak {p}_h^{k+1}( \varepsilon _h^u,\varepsilon _h^{\widehat{u}})$. Proceeding as in the proof of Proposition 1, it can be shown that $\varepsilon _h^u \in [\mathcal {P}_{\ell }^{k+1}(K)]^{\perp }$. Then, by equation (4b), the wanted inequality follows. This completes the proof.