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Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations I: An HDG\(_{k}\) Method

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Abstract

In our earlier work (Cockburn et al. in J Sci Comput 79(3):1777–1800, 2019), we approximated solutions of a general class of scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (interpolatory HDG) method. This method reduces the computational cost compared to standard HDG since the HDG matrices are assembled once before the time integration. Interpolatory HDG also achieves optimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reaction diffusion problems, and use the postprocessed approximate solution to evaluate the nonlinear term. We prove this simple change restores the superconvergence and keeps the computational advantages of the Interpolatory HDG method. We present numerical results to illustrate the convergence theory and the performance of the method.

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Acknowledgements

G. Chen is supported by National natural science Foundation of China (NSFC) under Grant Number 11801063 and China Postdoctoral Science Foundation under Grant Number 2018M633339. The research of Y. Zhang is partially supported by the US National Science Foundation (NSF) under Grant Number DMS-1619904.

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Authors and Affiliations

  1. College of Mathematics, Sichun University, Chengdu, China

    Gang Chen

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Bernardo Cockburn

  3. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

    John Singler

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Yangwen Zhang

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  1. Gang Chen
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Appendix A

Appendix A

Recall the steady state problem (24) from Sect. 3.2.4, which we repeat here for convenience: let \((\overline{\varvec{q}}_h,{\overline{u}}_h,\widehat{{{\overline{u}}}}_h)\in \varvec{V}_h\times W_h\times M_h\) be the solution of

$$\begin{aligned} {\mathscr {B}} (\overline{\varvec{q}}_h,{\overline{u}}_h,\widehat{\overline{u}}_h, \varvec{r}_h, v_h, {{\widehat{v}}}_h )= (f - \varPi _W u_t - F(u),v_h)_{{\mathcal {T}}_h}, \end{aligned}$$
(32)

for all \((\varvec{r}_h,v_h,{\widehat{v}}_h)\in \varvec{V}_h\times W_h\times M_h\). Since \( \varPi _W \) commutes with the time derivative, taking the partial derivative of (32) with respect to t shows \((\partial _t \overline{\varvec{q}}_h,\partial _t{\overline{u}}_h,\partial _t\widehat{{{\overline{u}}}}_h)\in \varvec{V}_h\times W_h\times M_h\) is the solution of

$$\begin{aligned} {\mathscr {B}} (\partial _t\overline{\varvec{q}}_h,\partial _t{\overline{u}}_h,\partial _t\widehat{{{\overline{u}}}}_h, \varvec{r}_h, v_h, {{\widehat{v}}}_h )&= (f_t - \varPi _W u_{tt} - F'(u)u_t,v_h)_{{\mathcal {T}}_h}, \end{aligned}$$

for all \((\varvec{r}_h,v_h,{\widehat{v}}_h)\in \varvec{V}_h\times W_h\times M_h\).

The proof of the following lemma is very similar to a proof in [13], hence we omit it here.

Lemma 6.1

For \(\varepsilon _h^{\overline{\varvec{q}}}=\varvec{\varPi }_{V}\varvec{q}- \overline{\varvec{q}}_h \), \( \varepsilon _h^{ {{\overline{u}}}}=\varPi _{W} {u}-{{\overline{u}}}_h \), and \( \varepsilon _h^{ \widehat{\overline{u}}}=P_M u-\widehat{{{\overline{u}}}}_h\), we have

$$\begin{aligned} {\mathscr {B}}(\varepsilon _h^{\overline{\varvec{q}}},\varepsilon _h^{\overline{u}},\varepsilon _h^{\widehat{{{\overline{u}}}}}; \varvec{r}_h, w_h, \widehat{v}_h) = (\varvec{\varPi }_V \varvec{q} - \varvec{q},\varvec{r}_h)_{{\mathcal {T}}_h}+ (\varPi _W u_t - u_t , v_h)_{{\mathcal {T}}_h}, \end{aligned}$$
(33)

for all \((\varvec{r}_h,v_h,{{\widehat{v}}}_h)\in \varvec{V}_h\times W_h\times M_h\).

The next step is the consideration of the dual problem (10), which we again repeat for convenience:

$$\begin{aligned} \begin{aligned}&\varvec{\varPhi }+\nabla \varPsi =0\qquad \qquad \text {in}\ \varOmega ,\\&\nabla \cdot \varvec{\varPhi }=\varTheta \qquad \quad ~~\text {in}\ \varOmega ,\\&\varPsi = 0\qquad \qquad \text {on}\ \partial \varOmega . \end{aligned} \end{aligned}$$
(34)

By the assumption at the beginning of Sect. 3, this boundary value problem admits the regularity estimate

$$\begin{aligned} \Vert \varvec{\varPhi }\Vert _{H^{1}(\varOmega )} + \Vert \varPsi \Vert _{H^{2}(\varOmega )} \le C \Vert \varTheta \Vert _{L^{2}(\varOmega )} \end{aligned}$$
(35)

for all \(\varTheta \in L^2(\varOmega )\).

Lemma 6.2

We have

$$\begin{aligned} \Vert \varepsilon ^{{{\overline{u}}}}_h\Vert _{{\mathcal {T}}_h}&\le Ch^{\min \{k,1\}} (\Vert \varvec{q} - \varvec{\varPi }_V \varvec{q}\Vert _{{\mathcal {T}}_h}+\Vert u_t - \varPi _W u_t\Vert _{{\mathcal {T}}_h}),\\ \Vert \varepsilon ^{\overline{\varvec{q}}}_h\Vert _{{\mathcal {T}}_h}&\le C (\Vert \varvec{q} - \varvec{\varPi }_V \varvec{q}\Vert _{{\mathcal {T}}_h} +\Vert u_t - \varPi _W u_t\Vert _{\mathcal T_h}). \end{aligned}$$

Proof

Let \(\varTheta = \varepsilon _h^{{{\overline{u}}}}\) in the dual problem (34), and take \((\varvec{r}_h,v_h,{{\widehat{v}}}_h) = (-\,\varvec{\varPi }_V\varvec{\varPhi }, \varPi _W\varPsi ,P_M \varPsi )\) in the definition of \( {\mathscr {B}} \) (16) to get

$$\begin{aligned}&{\mathscr {B}} \left( \varepsilon ^{\overline{\varvec{q}}}_h,\varepsilon ^{{{\overline{u}}}}_h,\varepsilon ^{\widehat{{{\overline{u}}}}}_h;-\varvec{\varPi }_V\varvec{\varPhi },\varPi _W\varPsi ,P_M\varPsi \right) \\&\quad =-\,\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPi }_V\varvec{\varPhi })_{{\mathcal {T}}_h}+(\varepsilon ^{{{\overline{u}}}}_h,\nabla \cdot \varvec{\varPi }_V\varvec{\varPhi }\right) _{{\mathcal {T}}_h}-\left\langle \varepsilon ^{\widehat{{{\overline{u}}}}}_h, \varvec{\varPi }_V\varvec{\varPhi }\cdot \varvec{n}\right\rangle _{\partial {\mathcal {T}}_h}+ \left( \nabla \cdot \varepsilon ^{\overline{\varvec{q}}}_h, \varPi _W\varPsi \right) _{{\mathcal {T}}_h}\\&\quad \quad -\,\left\langle \varepsilon ^{\overline{\varvec{q}}}_h\cdot \varvec{n}, P_M \varPsi \right\rangle _{\partial {{\mathcal {T}}_h}} + \left\langle \tau \left( \varepsilon ^{{{\overline{u}}}}_h-\varepsilon ^{\widehat{{{\overline{u}}}}}_h\right) , \varPi _W \varPsi - P_M\varPsi \right\rangle _{\partial {\mathcal {T}}_h}\\&\quad =-\,\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPhi }\right) _{{\mathcal {T}}_h}+\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi }\right) _{{\mathcal {T}}_h}+\left( \varepsilon ^{{{\overline{u}}}}_h,\nabla \cdot \varvec{\varPhi }\right) _{{\mathcal {T}}_h}-\left( \varepsilon ^{{{\overline{u}}}}_h,\nabla \cdot (\varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi })\right) _{{\mathcal {T}}_h}\\&\quad \quad +\,\langle \varepsilon ^{\widehat{{{\overline{u}}}}}_h, \left( \varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi }\right) \cdot \varvec{n}\rangle _{\partial {\mathcal {T}}_h} + \left( \nabla \cdot \varepsilon ^{\overline{\varvec{q}}}_h, \varPsi \right) _{{\mathcal {T}}_h} + \left( \nabla \cdot \varepsilon ^{\overline{\varvec{q}}}_h, \varPi _W \varPsi - \varPsi \right) _{{\mathcal {T}}_h}\\&\quad \quad -\left\langle \varepsilon ^{\overline{\varvec{q}}}_h\cdot \varvec{n}, \varPsi \right\rangle _{\partial {{\mathcal {T}}_h}} + \left\langle \tau \left( \varepsilon ^{{{\overline{u}}}}_h-\varepsilon ^{\widehat{{{\overline{u}}}}}_h\right) , \varPi _W \varPsi - P_M\varPsi \right\rangle _{\partial {\mathcal {T}}_h}\\&\quad =\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi }\right) _{{\mathcal {T}}_h} + \left\| \varepsilon ^{\overline{u}}_h\right\| _{{\mathcal {T}}_h}^2. \end{aligned}$$

On the other hand, take \((\varvec{r}_h,v_h,{{\widehat{v}}}_h) = (-\,\varvec{\varPi }_V\varvec{\varPhi }, \varPi _W\varPsi ,P_M \varPsi )\) in (33) to get

$$\begin{aligned} {\mathscr {B}} \left( \varepsilon ^{\overline{\varvec{q}}}_h,\varepsilon ^{\overline{u}}_h,\varepsilon ^{\widehat{\overline{u}}}_h;-\varvec{\varPi }_V\varvec{\varPhi },\varPi _W\varPsi ,P_M\varPsi \right) = (\varvec{q} - \varvec{\varPi }_V \varvec{q},\varvec{\varPi }_V\varvec{\varPhi })_{{\mathcal {T}}_h} + (\varPi _W u_t - u_t , \varPi _W \varPsi )_{{\mathcal {T}}_h}. \end{aligned}$$

Comparing the above two equalities gives

$$\begin{aligned} \Vert \varepsilon ^{{{\overline{u}}}}_h\Vert ^2_{{\mathcal {T}}_h}&=-\,\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi }\right) _{{\mathcal {T}}_h}+ (\varvec{q} - \varvec{\varPi }_V \varvec{q},\varvec{\varPi }_V\varvec{\varPhi })_{{\mathcal {T}}_h} + (\varPi _W u_t - u_t , \varPi _W \varPsi )_{{\mathcal {T}}_h} \\&=-\,\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi }\right) _{{\mathcal {T}}_h}+ (\varvec{q} - \varvec{\varPi }_V \varvec{q},\varvec{\varPi }_V\varvec{\varPhi } - \varvec{\varPhi })_{{\mathcal {T}}_h} \\&\quad +\, (\varvec{q} - \varvec{\varPi }_V \varvec{q},\varvec{\varPhi })_{{\mathcal {T}}_h} + (\varPi _W u_t - u_t , \varPi _W \varPsi )_{{\mathcal {T}}_h} \\&=-\,\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi }\right) _{{\mathcal {T}}_h}+ (\varvec{q} - \varvec{\varPi }_V \varvec{q},\varvec{\varPi }_V\varvec{\varPhi } - \varvec{\varPhi })_{{\mathcal {T}}_h} \\&\quad -\, (\varvec{q} - \varvec{\varPi }_V \varvec{q},\nabla \varPsi )_{{\mathcal {T}}_h} + (\varPi _W u_t - u_t , \varPi _W \varPsi )_{{\mathcal {T}}_h} \\&=-\,\left( \varepsilon ^{\overline{\varvec{q}}}_h,\varvec{\varPhi }- \varvec{\varPi }_V \varvec{\varPhi }\right) _{{\mathcal {T}}_h}+ (\varvec{q} - \varvec{\varPi }_V \varvec{q},\varvec{\varPi }_V\varvec{\varPhi } - \varvec{\varPhi })_{{\mathcal {T}}_h} \\&\quad -\, (\varvec{q} - \varvec{\varPi }_V \varvec{q},\nabla (\varPsi - \varPi _W\varPsi ))_{{\mathcal {T}}_h} + (\varPi _W u_t - u_t , \varPi _W \varPsi - \min \{k,1\}\varPi _0 \varPsi )_{{\mathcal {T}}_h}. \end{aligned}$$

Hence, by the regularity of the dual PDE (35), we have

$$\begin{aligned} \left\| \varepsilon ^{{{\overline{u}}}}_h\right\| ^2_{{\mathcal {T}}_h}&\le Ch^2 \left\| \varepsilon ^{\overline{\varvec{q}}}_h\right\| _{{\mathcal {T}}_h}^2 + Ch^{\min \{2k,2\}}\Vert \varvec{q} - \varvec{\varPi }_V \varvec{q}\Vert _{{\mathcal {T}}_h}^2 \nonumber \\&\quad + Ch^{\min \{2k,2\}}\Vert u_t - \varPi _W u_t\Vert _{{\mathcal {T}}_h}^2. \end{aligned}$$
(36)

Next, take \((\varvec{r}_h,v_h,{\widehat{v}}_h)=(\varepsilon _h^{\overline{\varvec{q}}},\varepsilon _h^{{{\overline{u}}}},\varepsilon _h^{\widehat{{{\overline{u}}}}})\) in (33) to obtain

$$\begin{aligned}&\left\| \varepsilon _h^{\overline{\varvec{q}}}\right\| ^2_{{\mathcal {T}}_h}+\left\langle \tau \left( \varepsilon _h^{{{\overline{u}}}} -\varepsilon _h^{\widehat{ {{\overline{u}}}}}\right) , \varepsilon _h^{{{\overline{u}}}} -\varepsilon _h^{\widehat{ {{\overline{u}}}}}\right\rangle _{\partial {{\mathcal {T}}_h}} \\&\quad = \left( \varvec{\varPi }_V {\varvec{q}} -\varvec{q}, \varepsilon _h^{\overline{\varvec{q}}}\right) _{{\mathcal {T}}_h} + \left( \varPi _W u_t - u_t,\varepsilon _h^{{{\overline{u}}}}\right) _{{\mathcal {T}}_h}\\&\quad \le C\Vert \varvec{\varPi }_V {\varvec{q}} -\varvec{q}\Vert _{{\mathcal {T}}_h}^2 + \frac{1}{2} \left\| \varepsilon _h^{\overline{\varvec{q}}}\right\| ^2_{{\mathcal {T}}_h} + 4C\Vert \varPi _W u_t - u_t\Vert _{{\mathcal {T}}_h}^2 +\frac{1}{4C} \left\| \varepsilon _h^{{{\overline{u}}}}\right\| _{{\mathcal {T}}_h}^2. \end{aligned}$$

This implies

$$\begin{aligned}&\left\| \varepsilon _h^{\overline{\varvec{q}}}\right\| ^2_{{\mathcal {T}}_h}+\left\langle \tau \left( \varepsilon _h^{{{\overline{u}}}} -\varepsilon _h^{\widehat{ {{\overline{u}}}}}\right) , \varepsilon _h^{{{\overline{u}}}} -\varepsilon _h^{\widehat{ {{\overline{u}}}}} \right\rangle _{\partial {{\mathcal {T}}_h}} \nonumber \\&\quad \le 2C\Vert \varvec{\varPi }_V {\varvec{q}} -\varvec{q}\Vert _{{\mathcal {T}}_h}^2 + 8C\Vert \varPi _W u_t - u_t\Vert _{{\mathcal {T}}_h}^2 +\frac{1}{2C} \Vert \varepsilon _h^{{{\overline{u}}}}\Vert _{{\mathcal {T}}_h}^2. \end{aligned}$$
(37)

Next, use \(h\le 1\) and substitute (37) into (36) to yield the result. \(\square \)

Following the same steps, we obtain the following result:

Lemma 6.3

We have

$$\begin{aligned} \Vert \partial _t(\varPi _W u - {{\overline{u}}}_h)\Vert _{{\mathcal {T}}_h}&\le Ch^{\min \{k,1\}} (\Vert \varvec{q}_t - \varvec{\varPi }_V \varvec{q}_t\Vert _{{\mathcal {T}}_h} +\Vert u_{tt} - \varPi _W u_{tt}\Vert _{{\mathcal {T}}_h}). \end{aligned}$$

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Chen, G., Cockburn, B., Singler, J. et al. Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations I: An HDG\(_{k}\) Method. J Sci Comput 81, 2188–2212 (2019). https://doi.org/10.1007/s10915-019-01081-3

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