Error estimation of a discontinuous Galerkin method for time fractional subdiffusion problems with nonsmooth data

Abstract

This paper is devoted to the numerical analysis of a piecewise constant discontinuous Galerkin method for time fractional subdiffusion problems. The regularity of weak solution is firstly established by using variational approach and Mittag-Leffler function. Then several optimal error estimates are derived with low regularity data. Finally, numerical experiments are conducted to verify the theoretical results.

Appendix A. Some properties of fractional calculus operators

Lemma 10

([35]) If $0<\alpha , \beta < \infty$, then

$$\begin{aligned} {{\,\mathrm{I}\,}}_{0+}^\beta {{\,\mathrm{I}\,}}_{0+}^\alpha v = {{\,\mathrm{I}\,}}_{0+}^{\beta +\alpha }v, \quad {{\,\mathrm{I}\,}}_{1-}^\beta {{\,\mathrm{I}\,}}_{1-}^\alpha v = {{\,\mathrm{I}\,}}_{1-}^{\beta +\alpha }v, \end{aligned}$$

for all $v\in L^1(0,1)$, and if $0< \alpha< \beta < \infty$, then

$$\begin{aligned} {{\,\mathrm{D}\,}}_{0+}^\beta {{\,\mathrm{I}\,}}_{0+}^\alpha v = {{\,\mathrm{D}\,}}_{0+}^{\beta -\alpha }v, \quad {{\,\mathrm{D}\,}}_{1-}^\beta {{\,\mathrm{I}\,}}_{1-}^\alpha v = {{\,\mathrm{D}\,}}_{1-}^{\beta -\alpha }v, \end{aligned}$$

for all $v\in L^1(0,1)$. Moreover, for all $v, w \in L^2(0,1)$,

$$\begin{aligned} \left\langle {{{\,\mathrm{I}\,}}_{0+}^\beta v,w} \right\rangle _{(0,1)} = \left\langle {v, {{\,\mathrm{I}\,}}_{1-}^\beta w} \right\rangle _{(0,1)}. \end{aligned}$$

Lemma 11

([3]) If $0< \gamma < 1/2$ and $v,w \in H^{\gamma }(0,T)$, then

$$\begin{aligned}&\left\langle {{{\,\mathrm{D}\,}}_{0+}^\gamma v, {{\,\mathrm{D}\,}}_{T-}^\gamma v} \right\rangle _{(0,T)} =\cos \gamma \pi \left|{v} \right|_{H^{\gamma }(0,T)}^2,\\&\left\langle {{{\,\mathrm{D}\,}}_{0+}^\gamma v, {{\,\mathrm{D}\,}}_{T-}^\gamma w} \right\rangle _{(0,T)} = \left\langle {{{\,\mathrm{D}\,}}_{0+}^{2\gamma } v, w} \right\rangle _{H^\gamma (0,T)} = \left\langle {{{\,\mathrm{D}\,}}_{T-}^{2\gamma } w, v} \right\rangle _{H^\gamma (0,T)},\\&\cos \gamma \pi \left\Vert {{{\,\mathrm{I}\,}}_{0+}^\gamma v} \right\Vert _{L^2(0,T)}^2 \leqslant \left\langle {{{\,\mathrm{I}\,}}_{0+}^\gamma v, {{\,\mathrm{I}\,}}_{T-}^\gamma v} \right\rangle _{(0,T)} \leqslant \sec \gamma \pi \left\Vert {{{\,\mathrm{I}\,}}_{0+}^\gamma v} \right\Vert _{L^2(0,T)}^2,\\&\cos \gamma \pi \left\Vert {{{\,\mathrm{D}\,}}_{0+}^\gamma v} \right\Vert _{L^2(0,T)}^2\leqslant \left\langle {{{\,\mathrm{D}\,}}_{0+}^\gamma v, {{\,\mathrm{D}\,}}_{T-}^\gamma v} \right\rangle _{(0,T)} \leqslant \sec \gamma \pi \left\Vert {{{\,\mathrm{D}\,}}_{0+}^\gamma v} \right\Vert _{L^2(0,T)}^2. \end{aligned}$$

Lemma 12

([22]) If $v \in {}_0H^\beta (0,1;\dot{H}^r(\varOmega )) \cap {}_0H^\gamma (0,1;\dot{H}^s(\varOmega ))$ with $\gamma ,\beta \geqslant 0$ and $s,r\in {\mathbb {R}}$, then for all $0< \theta < 1$,

$$\begin{aligned} \begin{aligned} {}&\left\Vert {v} \right\Vert _{ {}_0H^{\theta \beta + (1-\theta ) \gamma } (0,1;\dot{H}^{\theta r + (1-\theta )s}(\varOmega )) } \\ \leqslant {}&C_{\beta ,\gamma ,\theta } \left( \left\Vert {v} \right\Vert _{{}_0H^\beta (0,1;\dot{H}^r(\varOmega ))} + \left\Vert {v} \right\Vert _{{}_0H^\gamma (0,1;\dot{H}^s(\varOmega ))} \right) . \end{aligned} \end{aligned}$$

Similarly, if $v \in {}^0\!H^\beta (0,1;\dot{H}^r(\varOmega )) \cap {}^0\!H^\gamma (0,1;\dot{H}^s(\varOmega ))$ with $\gamma ,\beta \geqslant 0$ and $s,r\in {\mathbb {R}}$, then for all $0< \theta < 1$,

$$\begin{aligned} \begin{aligned} {}&\left\Vert {v} \right\Vert _{ {}^0\!H^{\theta \beta + (1-\theta ) \gamma } (0,1;\dot{H}^{\theta r + (1-\theta )s}(\varOmega )) } \\ \leqslant {}&C_{\beta ,\gamma ,\theta } \left( \left\Vert {v} \right\Vert _{{}^0\!H^\beta (0,1;\dot{H}^r(\varOmega ))} + \left\Vert {v} \right\Vert _{{}^0\!H^\gamma (0,1;\dot{H}^s(\varOmega ))} \right) . \end{aligned} \end{aligned}$$

Lemma 13

([22]) If $\beta \geqslant \gamma >0$, then

$$\begin{aligned} \left\Vert {{{\,\mathrm{D}\,}}_{T-}^{\gamma } v} \right\Vert _{{}^0\!H^{\beta -\gamma }(0,T)} \leqslant {}&C_{1}\left\Vert {v} \right\Vert _{{}^0\!H^\beta (0,T)}\quad \forall \,v \in {}^0\!H^\beta (0,T),\\ \left\Vert {{{\,\mathrm{D}\,}}_{0+}^{\gamma } v} \right\Vert _{{}_0H^{\beta -\gamma }(0,T)} \leqslant {}&C_{2} \left\Vert {v} \right\Vert _{{}_0H^\beta (0,T)} \quad \forall \,v \in {}_0H^\beta (0,T), \end{aligned}$$

where $C_1$ and $C_2$ depend only on $\gamma$ and $\beta$.

Lemma 14

([22]) If $\beta ,\gamma \geqslant 0$, then

$$\begin{aligned} C_{1}\left\Vert {v} \right\Vert _{{}^0\!H^\beta (0,T)} \leqslant \left\Vert {{{\,\mathrm{I}\,}}_{T-}^{\gamma } v} \right\Vert _{{}^0\!H^{\beta +\gamma }(0,T)} \leqslant {}&C_{2}\left\Vert {v} \right\Vert _{{}^0\!H^\beta (0,T)} \quad \forall \,v \in {}_0H^\beta (0,T),\\ C_{3}\left\Vert {v} \right\Vert _{{}_0H^\beta (0,T)} \leqslant \left\Vert {{{\,\mathrm{I}\,}}_{0+}^{\gamma } v} \right\Vert _{{}_0H^{\beta +\gamma }(0,T)} \leqslant {}&C_{4}\left\Vert {v} \right\Vert _{{}_0H^\beta (0,T)} \quad \forall \,v \in {}_0H^\beta (0,T). \end{aligned}$$

where $C_1,\,C_2,\,C_3$ and $C_4$ depend only on $\gamma$ and $\beta$.

Lemma 15

([22]) If $0< \gamma <1/2$, then for all $v \in {}_0H^1(0,1)$,

$$\begin{aligned} \left\Vert {v} \right\Vert _{C[0,1]} \leqslant C_{\gamma } \left\Vert {v} \right\Vert _{{}_0H^1(0,1)}^{(1/2-\gamma )/(1-\gamma )} \left\Vert {v} \right\Vert _{{}_0H^\gamma (0,1)}^{1/(2-2\gamma )}. \end{aligned}$$

Moreover, if $v\in {}_0H^{\gamma }(0,1)$ with $1/2<\gamma \leqslant 1$, then for all $0<\epsilon \leqslant 1$,

$$\begin{aligned} \left\Vert {v} \right\Vert _{C[0,1]}\leqslant \frac{C_{\gamma }}{\sqrt{\epsilon }} \left\Vert {v} \right\Vert _{{}_0H^{1/2}(0,1)}^{1-\epsilon } \left\Vert {v} \right\Vert _{{}_0H^{\gamma }(0,1)}^{\epsilon }. \end{aligned}$$