Analysis of a full-discrete two-grid discontinuous Galerkin method for nonlinear parabolic equations

Abstract

A full-discrete two-grid discontinuous Galerkin approximation for nonlinear parabolic problems is proposed. The $L^2$-norm error analysis of the two-grid method is carried out. The analysis shows that our algorithm will achieve an asymptotically optimal approximation as long as mesh sizes satisfy $h = O(H^2)$, where H and h are the size of the coarse mesh and the fine mesh, respectively. Numerical examples are presented to test the efficiency of our algorithm.

1 Introduction

Consider a nonlinear parabolic problem:

$$\begin{aligned} \left\{ \begin{array}{lll} u_t - \nabla \cdot (a(u) \nabla u) = f(u), \quad (x, t) \in \Omega \times (0, T],\\ u(x,t) = 0, \quad (x, t) \in \partial \Omega \times (0, T],\\ u(x,0) = u_0 (x), \quad x \in \Omega , \end{array} \right. \end{aligned}$$

(1.1)

where $u_t = \frac{\partial u}{\partial t}$, $\Omega \subset R^2$ is a bounded, convex, polygonal domain with the smooth boundary $\partial \Omega$. The equations describe the diffusion process of a chemical species of concentration u in a porous medium with a nonlinear source term f.

Suppose that a(u) is a bounded smooth function, $0 < \alpha \le a(u) \le \beta$. And, a(u) and f(u) are continuously differentiable and $|a'(u)| \le M$, $|f'(u)| + |f''(u)| \le M$ hold, where M is a positive constant. It is also assumed that $u_0$ is smooth enough to ensure that the model problem (1.1) has a unique solution.

Discontinuous Galerkin methods (DG) have become very popular for solving partial differential equations (see Chen and Chen 2004; Rivière and Wheeler 2002; Romkes et al. 2003; Sun 2003; Sun and Wheeler 2005; Yang and Chen 2006, 2010, 2011, 2012; Yang et al. 2013; Yang and Xiong 2013) because of their attractive properties, such as the local mass conservation, the better convergence behavior, the flexibility in handling of the complicated geometry and the mesh adaptation.

The two-grid method, firstly introduced as a discretization method by Xu (1994, 1996), is effective for solving the nonlinear problem. For the nonlinear parabolic problem (1.1), Chen and Liu (2012) have constructed a two-grid finite element method. In Chen and Liu (2010), Chen et al. (2003, 2009), two-grid finite volume element methods for semilinear and nonlinear parabolic problems are analysed. Chen and Li (2009), Chen et al. (2007) have studied two-grid methods with expanded mixed finite element solutions for nonlinear parabolic problems, which found that the two-grid scheme based on the expanded mixed finite element method can keep the same convergence order as the standard expanded mixed finite element method and cost much less work. Two-grid methods with finite difference and mixed finite element are studied by Dawson et al. (1998), Dawson and Wheeler (1994), which has established error estimates for two-grid approximation schemes. Error estimates for discontinuous Galerkin method for the nonlinear parabolic problem (1.1) are given in Ohm et al. (2013), Riviere and Wheeler (2000), Song et al. (2013). In Bi and Ginting (2011), a two-grid discontinuous Galerkin method is presented for quasilinear elliptic problems. But there is little literature about the error analysis of a two-grid discontinuous Galerkin method for the nonlinear parabolic problem. In Yang (2015), error estimates of a semi-discrete two-grid discontinuous Galerkin method are given for nonlinear parabolic equations with linear source term, and a full-discrete two-grid DG approximation is proposed without analysis. In this paper, the error analysis of a full-discrete two-grid discontinuous Galerkin method for nonlinear parabolic equations (1.1) has been made. We carry out the stability analysis of the discrete solution and present the error estimate in $H^1$ norm of the discontinuous Galerkin method and the estimate in $L^2$ norm of the two-grid discontinuous Galerkin algorithm.

The paper is organized as follows. In Sect. 2, we introduce a full-discrete discontinuous Galerkin method and derive the stability of the discrete solution to the nonlinear problem under consideration. In Sect. 3, error estimates of the discontinuous Galerkin approximation are presented. Section 4 displays a two-grid discontinuous Galerkin method and its error analysis. The numerical experiments are presented in the fifth section. In the last part, the conclusions are given.

2 A full-discrete discontinuous Galerkin method

2.1 Notation

Let $\mathcal {T}_{h} = \{E_1, E_2, \ldots , E_{N_h}\}$ be a quasi-uniform partition of $\Omega$, with E being a triangle or quadrilateral with the diameter $h_E$. Denote by $\Gamma _{h}$ the set of all interior edges of $\mathcal {T}_{h}$ and by ${\varvec{n}}$ the outward unit normal vector on each edge $\gamma \in \Gamma _{h} \cup \partial \Omega$. Let $h = \max \limits _{E \in \mathcal {T}_{h}} {h_{E}}$ be the maximal element diameter over all elements.

For $s \ge 0$, we define the following broken Sobolev space

$$\begin{aligned} H^{s}(\mathcal {T}_{h})=\{v \in L^{2}(\Omega ): v|_{E} \in H^{s}(E), E \in \mathcal {T}_{h}\}. \end{aligned}$$

For $E_i \in \mathcal {T}_{h}$, $E_j \in \mathcal {T}_{h}$ and $v \in H^{s}(\mathcal {T}_{h}), s > 1/2$, we define the average $\{v\}$ of v on $\gamma = \partial E_{i} \cap \partial E_{j}$ with ${\varvec{n}}$ exterior to $E_{i}$ and the jump [v] of v across $\gamma$ as follows.

$$\begin{aligned} \left\{ v\right\} = \frac{1}{2}\Big ( (v|_{E_{i}})|_{\gamma } + (v|_{E_{j}})|_{\gamma } \Big ),\quad \left[ v \right] = (v|_{E_{i}})|_{\gamma } - (v|_{E_{j}})|_{\gamma }. \end{aligned}$$

The $L^2$ inner product is denoted by $(\cdot , \cdot )$. For $m \ge 0, 1 \le p < \infty$ on the element E, we use the standard Sobolev space $W^{m, p} (E)$ with a norm $\Vert \cdot \Vert _{m,p,E}$. For $p = 2$, we define $\Vert \cdot \Vert _{m, E} = \Vert \cdot \Vert _{m, 2, E}$, $\Vert \cdot \Vert _{\infty , E} = \Vert \cdot \Vert _{L^{\infty } (E)}$ and $\Vert \cdot \Vert _{E} = \Vert \cdot \Vert _{0, 2, E}$. The same symbols are used for the edge $\gamma$. We use the following broken norms:

$$\begin{aligned} \Vert v\Vert= & {} \left( \sum \limits _{E \in \mathcal {T}_{h}} \Vert v\Vert _{E}^2 \right) ^{1/2},\\ \Vert v\Vert _1= & {} \left( \sum \limits _{E \in \mathcal {T}_{h}} \Vert v\Vert _{1, E}^2 + J^\sigma (v, v) \right) ^{1/2}, \end{aligned}$$

where $J^\sigma (u, v) = \sum \nolimits _{\gamma \in \Gamma _{h}} \frac{ \sigma _{\gamma }}{h_{\gamma }} \int _{\gamma } [u] [v] d s$ is the interior penalty term, $\sigma$ is a positive real function that takes constant value $\sigma _\gamma$ on the edge $\gamma$ and is bounded, $h_\gamma$ is the size of $\gamma$.

We shall use the following discontinuous finite element space:

$$\begin{aligned} V_h = \{v \in L^{2}(\Omega ): v|_{E} \in P_{r}(E), E \in \mathcal {T}_{h}\}, \end{aligned}$$

where $P_{r}(E)$ denotes the space of polynomials of total degree less than or equal to r on E.

Like Yang and Chen (2010), throughout the paper, we shall use $K, K_i$ $(i = 1,2,\ldots )$ to denote generic positive constants which are independent of h, but might depend on the solution of PDEs with different values at different occurrences. Let $\varepsilon$ denote a fixed positive constant that can be chosen arbitrarily small.

2.2 The full-discrete discontinuous Galerkin method

To solve the problem (1.1), the following numerical scheme is considered, i.e., the discontinuous Galerkin method is used for space variables and the backward Euler scheme is used for time discretisation.

First, we introduce the trilinear form $B(\omega ; u, v)$ and $B_\lambda (\omega ; u, v)$:

$$\begin{aligned} B(\omega ; u, v)= & {} \sum \limits _{E \in \mathcal {T}_{h}}\int _{E} a (\omega ) \nabla u \cdot \nabla v d x - \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{ a (\omega ) \nabla u \cdot {\varvec{n}} \} [v] d s \\{} & {} - \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{a (\omega ) \nabla v \cdot {\varvec{n}} \} [u] d s + J^\sigma (u, v),\\ B_\lambda (\omega ; u, v)= & {} B(\omega ; u, v) + \lambda (u, v) \end{aligned}$$

where $\lambda$ is a positive constant.

The semi-discrete discontinuous Galerkin approximating $u_h(\cdot , t) \in V_h$ to the solution u of the Eq. (1.1) is written as follows:

$$\begin{aligned} \begin{array}{lll} \left( \frac{\partial u_h}{\partial t},v\right) + B(u_h; u_h, v) = (f(u_h), v), \quad v \in V_h, \quad t >0,\\ u_h(\cdot , 0) = P_h u_0, \end{array} \end{aligned}$$

where $P_h u_0$ is an appropriate projection of $u_0$ to be defined later.

Let N be a positive integer. $\Delta t = \frac{T}{N}, t^n = n \Delta t, \quad n = 0, 1, \ldots , N$. Denote $u^n_h = u_h(t^n)$. A full-discrete discontinuous Galerkin approximation to (1.1) is to find $u^n_h \in V_h$ for $n = 0, 1, \ldots , N$ such that

$$\begin{aligned} \begin{array}{lll} \left( \frac{u^n_h - u^{n-1}_h}{\Delta t},v_h\right) + B(u_h^n; u_h^n, v_h) = (f(u_h^n), v_h), \quad v_h \in V_h, \\ u_h^0 = P_h u_0. \end{array} \end{aligned}$$

(2.1)

The full-discrete formulation (2.1) can be rewritten as

$$\begin{aligned} \begin{array}{lll} (u^n_h,v_h) + \Delta t B(u_h^n; u_h^n, v_h) = (u^{n-1}_h + \Delta t f(u_h^n), v_h), \quad v_h \in V_h, \\ u_h^0 = P_h u_0. \end{array} \end{aligned}$$

By using the result in Riviere and Wheeler (2000), it is known that there exists a unique solution $u^n_h$ for $n = 0, 1, \ldots , N$.

The Gronwall’s lemma in Brenner and Scott (1994) will be recalled.

Lemma 2.1

Let $C_0$ and $b_k$, $c_k$, $d_k$ and $l_k$ ($k \ge 0$) be non-negative numbers such that

$$\begin{aligned} l_n + \Delta t \sum \limits _{k=0}^n b_k \le \Delta t \sum \limits _{k=0}^n d_k l_k + \Delta \sum \limits _{k=0}^n c_k + C_0, \quad n \ge 1, \end{aligned}$$

then,

$$\begin{aligned} l_n + \Delta t \sum \limits _{k=0}^n b_k \le \left( \Delta t \sum \limits _{k=0}^n c_k + C_0 \right) \exp \left( \Delta t \sum \limits _{k=0}^n d_k \right) , \quad n \ge 1. \end{aligned}$$

For the trilinear form $B_\lambda (\cdot ; \cdot , \cdot )$, referring to Ohm et al. (2013), we have

Lemma 2.2

There exists a positive constant K such that for sufficiently large $\sigma$,

$$\begin{aligned} |B_\lambda (\omega ; u, v)| \le K \Vert u\Vert _1 \Vert v\Vert _1, \forall \omega , v \in V_h. \end{aligned}$$

Lemma 2.3

For $\lambda > 0$, there exists a positive constant $\alpha _0$ such that for sufficiently large $\sigma$,

$$\begin{aligned} \alpha _0 \Vert v\Vert _1^2 \le B_\lambda (\omega ; v, v), \forall \omega , v \in V_h. \end{aligned}$$

Remark 1

By Lemmas 2.2 and 2.3, there exists a unique $\widetilde{u}$ satisfying

$$\begin{aligned} \begin{array}{lll} B_\lambda (\omega ; u - \widetilde{u}, v) = 0, \forall \omega , v \in V_h. \end{array} \end{aligned}$$

(2.2)

Now, we state the stability of the full-discrete discontinuous Galerkin approximation (2.1) for the problem (1.1).

Theorem 2.1

Let $u^n$ and $u_h^n$ be the solutions of the problem (1.1) and the full-discrete discontinuous Galerkin approximation (2.1), respectively. If $u_h^0 = P_h u_0$, $f \in L^2(\Omega )$, then for $1 \le l \le N$,

$$\begin{aligned} \begin{array}{lll} \Vert u^l_h\Vert ^2 + \alpha _0 \sum \limits _{n=1}^{l} \Vert u_h^n \Vert _1^2 \Delta t \le K \Vert u^0_h\Vert ^2 + K \sum \limits _{n=1}^{l} \Vert f(u_h^n)\Vert ^2 \Delta t, \end{array} \end{aligned}$$

where $\alpha _0$ is defined in Lemma 2.3.

Proof

Taking $v = u_h^n$ in Eq. (2.1), we have

$$\begin{aligned} \left( \frac{u^n_h - u^{n-1}_h}{\Delta t}, u_h^n\right) + B_\lambda (u_h^n; u_h^n, u_h^n) = (f(u_h^n), u_h^n) + \lambda (u_h^n, u_h^n). \end{aligned}$$

(2.3)

Since

$$\begin{aligned} \left( \frac{u^n_h - u^{n-1}_h}{\Delta t}, u_h^n\right)= & {} \frac{1}{2 \Delta t} (u^n_h - u^{n-1}_h, u_h^n + u^{n-1}_h) + \frac{1}{2 \Delta t} (u^n_h - u^{n-1}_h, u_h^n - u^{n-1}_h)\nonumber \\\ge & {} \frac{1}{2 \Delta t} (u^n_h - u^{n-1}_h, u_h^n + u^{n-1}_h) \nonumber \\= & {} \frac{1}{2 \Delta t} (\Vert u^n_h\Vert ^2 - \Vert u^{n-1}_h\Vert ^2), \end{aligned}$$

(2.4)

Substituting (2.4) into (2.3), multiplying by $2 \Delta t$, summing (2.3) from $n=1$ to $n=l$ $(1 \le l \le N)$ and using Lemma 2.3 and Young’s inequality, we get

$$\begin{aligned} \Vert u^l_h\Vert ^2 - \Vert u^0_h\Vert ^2 + 2 \alpha _0 \sum \limits _{n=1}^{l} \Vert u_h^n \Vert _1^2 \Delta t\le & {} 2 \sum \limits _{n=1}^{l} (\Vert f(u_h^n)\Vert \Vert u_h^n\Vert + \Vert u_h^n\Vert ^2) \Delta t\\\le & {} K \sum \limits _{n=1}^{l} \Vert f(u_h^n)\Vert ^2 \Delta t + \varepsilon \sum \limits _{n=1}^{l} \Vert u_h^n \Vert ^2 \Delta t. \end{aligned}$$

Applying the Gronwall’s lemma, we obtain the desired estimate.

3 Error estimates of the discontinuous Galerkin approximation

3.1 The approximation properties

First, we shall state some approximation properties whose proof can be found in Babus̆ka and Suri (1987a, 1987b).

Lemma 3.1

(approximation properties) For $E \in {\mathcal {T}}_{h}$, $\vartheta \in H^{s}({\mathcal {T}}_{h})$, there exists a sequence $\widehat{\vartheta } \in P_{r}(E),\quad r = 1, 2, \ldots$, and there exists a positive constant K depending on s but independent of r, h, such that for $0 \le l \le s$, $1\le m < \infty$, and $\mu = \min (r + 1, s)$,

$$\begin{aligned} \Vert \vartheta - \widehat{\vartheta }\Vert _{l, m, E}\le & {} K \frac{h_E^{\mu -l}}{r^{s-l}}\Vert \vartheta \Vert _{s, m, E} \quad s \ge 0,\end{aligned}$$

(3.1)

$$\begin{aligned} \Vert \vartheta - \widehat{\vartheta }\Vert _{\delta , \partial E}\le & {} K \frac{h_E^{\mu -\delta -1/2}}{r^{s-\delta -1/2}}\Vert \vartheta \Vert _{s, E} \quad s > \frac{1}{2}+ \delta , \quad \delta = 0, 1. \end{aligned}$$

(3.2)

Moreover, for $\gamma = \partial E_{i} \cap \partial E_{j}$,

$$\begin{aligned} \Vert \nabla \widehat{\vartheta }\Vert _{\infty , \gamma } \le K \Vert \nabla \vartheta \Vert _{\infty , E_{i} \cup E_{j}}. \end{aligned}$$

(3.3)

Denote $\eta ^n = u^n - \widetilde{u}^n$, $\theta ^n = \widehat{u}^n - \widetilde{u}^n$, $\xi ^n = \widetilde{u}^n - u^n_h$. According to Theorem 4.1 in Ohm et al. (2013), we have

Lemma 3.2

For $r, s \ge 2$, there exists a constant K satisfying

$$\begin{aligned}{} & {} \Vert \eta ^n\Vert \le K h^{\mu } \Vert u\Vert _{s},\\{} & {} \Vert \eta ^n\Vert _1 \le K h^{\mu - 1} \Vert u\Vert _{s},\\{} & {} \Vert \eta _t^n\Vert \le K h^{\mu } (\Vert u\Vert _{s} + \Vert u_t\Vert _{s}),\\{} & {} \Vert \eta _t^n\Vert _1 \le K h^{\mu - 1} (\Vert u\Vert _{s} + \Vert u_t\Vert _{s}), \end{aligned}$$

where $\mu = \min (r + 1, s)$.

According to Lemma 4.4 in Ohm et al. (2013), we get

Lemma 3.3

For $r, s \ge 2$, $\gamma = \partial E_{i} \cap \partial E_{j}$, there exists a constant K satisfying

$$\begin{aligned}{} & {} \Vert \nabla \widetilde{u}^n\Vert _{\infty } \le K \Vert u\Vert _{s},\\{} & {} \Vert \nabla \widetilde{u}^n\Vert _{\infty , \gamma } \le K \Vert u\Vert _{s}. \end{aligned}$$

In order to proceed the error analysis, the following Trace theorem and trace inequalities (Romkes et al. 2003; Sun 2003) are needed.

Lemma 3.4

(Trace theorem) Suppose that $\Omega$ has a Lipschitz boundary, p is a real number with $1 \le p < \infty$. Then there is a constant K, s.t.

$$\begin{aligned} \Vert v\Vert _{L^{p} (\partial \Omega )} \le K \Vert v\Vert _{L^{p} (\Omega )}^{1 - \frac{1}{p}} \Vert v\Vert _{W^{1, p} (\Omega )}^{\frac{1}{p}}, \quad \forall v \in W^{1, p} (\Omega ). \end{aligned}$$

Lemma 3.5

For each $E \in \mathcal {T}_{h}$, let $\gamma$ be an edge of E. Then there exists a positive constant K such that the following trace inequalities are valid:

$$\begin{aligned}{} & {} \Vert v\Vert _{\gamma }^{2} \le K (h_{E}^{-1} \Vert v\Vert _{E}^2 + h_{E} |v|_{1, E}^{2} ), \forall v \in H^{1}(E),\\{} & {} \Vert \nabla v \cdot {\varvec{n}}\Vert _{\gamma }^{2} \le K (h_{E}^{-1} |v|_{1, E}^2 + h_{E} |v|_{2, E}^{2} ), \forall v \in H^{2}(E). \end{aligned}$$

3.2 The error analysis

Here, we give the error estimate in $H^1$ norm for the problem (1.1) of the full-discrete discontinuous Galerkin approximation (2.1).

Theorem 3.1

Let $u^n$ and $u_h^n$ be the solution of the problem (1.1) and the full-discrete discontinuous Galerkin approximation (2.1), respectively. Let $\Delta t = O(h^{r+1})$. If $u_h^0 = P_h u_0$, then

$$\begin{aligned} \begin{array}{lll} \Vert u^n - u_h^n \Vert _1 \le K (h^{r} + \Delta t), \end{array} \end{aligned}$$

(3.4)

where the positive constant K may depend on u and $\frac{\partial u}{\partial t}$.

Proof

For $v \in V_h$, we get the error equation from (1.1) and (2.1)

$$\begin{aligned}{} & {} \left( u^n_t - \frac{u^n_h - u^{n-1}_h}{\Delta t}, v\right) + B_\lambda (u^n; u^n, v) - B_\lambda (u^n_h; u^n_h, v)\nonumber \\ {}{} & {} \quad = (f(u^n) - f(u^n_h), v) + \lambda (u^n - u^n_h, v). \end{aligned}$$

(3.5)

Denote $\partial _t \xi ^n = \frac{\xi ^n - \xi ^{n-1}}{\Delta t}$. Choosing $v = \partial _t \xi ^n$ in Eq. (3.5) and using the notations of $\xi ^n$ and $\eta ^n$, we get

$$\begin{aligned} (\partial _t \xi ^n, \partial _t \xi ^n) + B_\lambda (u^n_h; \xi ^n, \partial _t \xi ^n)= & {} (\partial _t u^n - u^n_t, \partial _t \xi ^n) - (\partial _t \eta ^n, \partial _t \xi ^n) + B_\lambda (u^n_h; \widetilde{u}^n, \partial _t \xi ^n) \nonumber \\{} & {} - B_\lambda (u^n; \widetilde{u}^n, \partial _t \xi ^n) - B_\lambda (u^n; \eta ^n, \partial _t \xi ^n) \nonumber \\{} & {} + (f(u^n) - f(u^n_h), \partial _t \xi ^n) + \lambda (\xi ^n + \eta ^n, \partial _t \xi ^n). \end{aligned}$$

(3.6)

Note that

$$\begin{aligned} B_\lambda (u^n_h; \xi ^n, \partial _t \xi ^n)= & {} \frac{1}{2 \Delta t} \left( B_\lambda (u^n_h; \xi ^n + \xi ^{n-1}, \xi ^n - \xi ^{n-1}) + B_\lambda (u^n_h; \xi ^n - \xi ^{n-1}, \xi ^n - \xi ^{n-1}) \right) \\\ge & {} \frac{1}{2 \Delta t} B_\lambda (u^n_h; \xi ^n + \xi ^{n-1}, \xi ^n - \xi ^{n-1})\\= & {} \frac{1}{2 \Delta t} \left( B_\lambda (u^n_h; \xi ^n, \xi ^n) - B_\lambda (u^n_h; \xi ^{n-1}, \xi ^{n-1}) \right) \\{} & {} - \frac{1}{2} \left( B_\lambda (u^n_h; \partial _t \xi ^n, \xi ^n) - B_\lambda (u^n_h; \xi ^n, \partial _t \xi ^n) \right) . \end{aligned}$$

By the definition of the trilinear form $B_\lambda (\cdot ; \cdot , \cdot )$, we have

$$\begin{aligned} \begin{array}{lll} |B_\lambda (u^n_h; \partial _t \xi ^n, \xi ^n) - B_\lambda (u^n_h; \xi ^n, \partial _t \xi ^n)|\\ \quad = |- \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{ a (u^n_h) \nabla \partial _t \xi ^n \cdot {\varvec{n}} \} [\xi ^n] d s - \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{a (u^n_h) \nabla \xi ^n \cdot {\varvec{n}} \} [\partial _t \xi ^n] d s\\ \qquad + \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{ a (u^n_h) \nabla \xi ^n \cdot {\varvec{n}} \} [\partial _t \xi ^n] d s + \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{a (u^n_h) \nabla \partial _t \xi ^n \cdot {\varvec{n}} \} [\xi ^n] d s|\\ \quad = 0. \end{array} \end{aligned}$$

Then,

$$\begin{aligned} \begin{array}{lll} B_\lambda (u^n_h; \xi ^n, \partial _t \xi ^n) = \frac{1}{2 \Delta t} \left( B_\lambda (u^n_h; \xi ^n, \xi ^n) - B_\lambda (u^n_h; \xi ^{n-1}, \xi ^{n-1}) \right) . \end{array} \end{aligned}$$

(3.7)

According to Lemma 2.3, we have

$$\begin{aligned} \begin{array}{lll} \sum \limits _{n=1}^{l} B_\lambda (u^n_h; \xi ^n, \partial _t \xi ^n) = \frac{1}{2 \Delta t} B_\lambda (u^l_h; \xi ^l, \xi ^l) \ge \frac{\alpha _0}{2 \Delta t} \Vert \xi ^l\Vert _1^2. \end{array} \end{aligned}$$

(3.8)

To estimate the items on the right hand side of (3.6), we proceed as follows. Applying Cauchy–Schwarz inequality, we have

$$\begin{aligned} |(\partial _t u^n - u^n_t, \partial _t \xi ^n) \Delta t |\le & {} \Vert \partial _t u^n - u^n_t\Vert \cdot \Vert \partial _t \xi ^n\Vert \Delta t\\\le & {} K ((\int _{t^{n-1}}^{t^n} \Vert u^n_{tt}\Vert d t)^2 + \Vert \partial _t \xi ^n\Vert ^2) \Delta t\\\le & {} K \left( \int _{t^{n-1}}^{t^n} \Vert u^n_{tt}\Vert ^2 d t \right) (\Delta t)^2 + \varepsilon \Vert \partial _t \xi ^n\Vert ^2 \Delta t,\\ |- (\partial _t \eta ^n, \partial _t \xi ^n) \Delta t|\le & {} \Vert \partial _t \eta ^n\Vert \cdot \Vert \partial _t \xi ^n\Vert \Delta t \\\le & {} K \int _{t^{n-1}}^{t^n} \Vert \eta ^n_{t}\Vert ^2 d t + \varepsilon \Vert \partial _t \xi ^n\Vert ^2 \Delta t. \end{aligned}$$

Using (2.2), it is easy to find that

$$\begin{aligned}{} & {} B_\lambda (u^n_h; \widetilde{u}^n, \partial _t \xi ^n) - B_\lambda (u^n; \widetilde{u}^n, \partial _t \xi ^n) - B_\lambda (u^n; \eta ^n, \partial _t \xi ^n) = B_\lambda (u^n_h; \widetilde{u}^n, \partial _t \xi ^n) - B_\lambda (u^n; \widetilde{u}^n, \partial _t \xi ^n)\nonumber \\{} & {} \quad = \sum \limits _{E \in \mathcal {T}_{h}}\int _{E} (a(u^n_h)- a(u^n)) \nabla \widetilde{u}^n \cdot \nabla \partial _t \xi ^n d x - \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{ (a (u^n_h) - a (u^n)) \nabla \widetilde{u}^n \cdot {\varvec{n}} \} [\partial _t \xi ^n] d s\nonumber \\{} & {} \qquad - \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{(a (u^n_h) - a (u^n)) \nabla \partial _t \xi ^n \cdot {\varvec{n}} \} [\widetilde{u}^n] d s\nonumber \\{} & {} \quad \equiv J_1 + J_2 + J_3. \end{aligned}$$

(3.9)

Note that $a(u^n) - a(u^n_h) = a'(\widehat{u}^n) (u^n - u^n_h)$ for some $\widehat{u}^n$ between $u^n$ and $u^n_h$ by the mean value theorem. For $J_1, J_2$, using Lemma 3.3, we have

$$\begin{aligned} |J_1|\le & {} K \Vert \nabla \widetilde{u}^n\Vert _{\infty } \Vert a(u^n_h)- a(u^n)\Vert \Vert \nabla \partial _t \xi ^n\Vert \\\le & {} K |\nabla \widetilde{u}^n|_{\infty } \Vert u^n_h - u^n\Vert \cdot \Vert \nabla \partial _t \xi ^n\Vert \\\le & {} K h^{-1} \Vert u^n_h - u^n\Vert \cdot \Vert \partial _t \xi ^n\Vert \\\le & {} K h^{-2} \Vert u^n_h - u^n\Vert ^2 + \varepsilon \Vert \partial _t \xi ^n\Vert ^2\\\le & {} K (h^r + h^{-1} \Delta t)^2 + \varepsilon \Vert \partial _t \xi ^n\Vert ^2,\\ |J_2|\le & {} \sum \limits _{\gamma \in \Gamma _{h}} K \Vert \nabla \widetilde{u}^n\Vert _{\infty , \gamma } \Vert \{u^n_h - u^n\}\Vert _{\gamma } \Vert [\partial _t \xi ^n]\Vert _{\gamma }\\\le & {} K \sum \limits _{\gamma \in \Gamma _{h}} \Vert \{u^n_h - u^n\}\Vert _{\gamma } \Vert [\partial _t \xi ^n]\Vert _{\gamma }\\\le & {} K h^{-1} \Vert u^n_h - u^n\Vert \cdot \Vert \partial _t \xi ^n\Vert \\\le & {} K (h^r + h^{-1} \Delta t)^2 + \varepsilon \Vert \partial _t \xi ^n\Vert ^2. \end{aligned}$$

where Lemmas 3.1 and 3.5, and the Cauchy–Schwarz inequality are used.

For $J_3$, using the inverse inequality, we get

$$\begin{aligned} |J_3|= & {} |\sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{(a (u^n_h) - a (u^n)) \nabla \partial _t \xi ^n \cdot {\varvec{n}} \} [\eta ^n]|\\\le & {} K \sum \limits _{\gamma \in \Gamma _{h}} \Vert \nabla \partial _t \xi ^n \Vert _{\infty , \gamma } \Vert \{u^n_h - u^n\}\Vert _{\gamma } \cdot \Vert [\eta ^n]\Vert _{\gamma }\\\le & {} K \sum \limits _{\gamma \in \Gamma _{h}} h^{-1/2} \Vert \nabla \partial _t \xi ^n\Vert _{\gamma } \Vert \{u^n_h - u^n\}\Vert _{\gamma } \cdot \Vert [\eta ^n]\Vert _{\gamma }\\\le & {} K \sum \limits _{E \in \mathcal {T}_{h}} h^{-1}\Vert \nabla \partial _t \xi ^n\Vert _{E} h^{-1/2} \Vert \{u^n_h - u^n\}\Vert _{E} \cdot (h^{-1/2} \Vert \eta ^n\Vert _{E} + h^{1/2} \Vert \nabla \eta ^n\Vert _{E})\\\le & {} K h^{-2}\Vert \nabla \partial _t \xi ^n\Vert (h^{r+1} + \Delta t) h^2 \Vert u^n\Vert _{2} \\\le & {} K h^{-1}\Vert \partial _t \xi ^n\Vert (h^{r+1} + \Delta t)\\\le & {} K (h^r + h^{-1} \Delta t)^2 + \varepsilon \Vert \partial _t \xi ^n\Vert ^2. \end{aligned}$$

Using the boundness of $f'(\cdot )$, we have

$$\begin{aligned} |(f(u^n) - f(u^n_h), \partial _t \xi ^n)|\le & {} K \Vert u^n - u^n_h\Vert \cdot \Vert \partial _t \xi ^n\Vert \\\le & {} K \Vert u^n - u^n_h\Vert ^2 + \varepsilon \Vert \partial _t \xi ^n\Vert ^2 \\\le & {} K (\Vert \xi ^n\Vert ^2 + \Vert \eta ^n\Vert ^2) + \varepsilon \Vert \partial _t \xi ^n\Vert ^2, \end{aligned}$$

where the mean value theorem $f(u^n) - f(u^n_h) = f'(\widehat{u}^n) (u^n - u^n_h)$ is used for some $\widehat{u}^n$ between $u^n$ and $u^n_h$.

For the last item on the right hand side of (3.6), we have

$$\begin{aligned} \lambda (\xi ^n + \eta ^n, \partial _t \xi ^n) \le K (\Vert \xi ^n\Vert ^2 + \Vert \eta ^n\Vert ^2) + \varepsilon \Vert \partial _t \xi ^n\Vert ^2. \end{aligned}$$

Multiplying by $2 \Delta t$ and summing Eq. (3.6) from $n=1$ to $n=l \quad (1 \le l \le N)$, collecting all the above estimates, we have

$$\begin{aligned} \alpha _0 \Vert \xi ^l\Vert _1^2 + 2 \sum \limits _{n=1}^{l} \Vert \partial _t \xi ^n\Vert ^2 \Delta t\le & {} K \left( \sum \limits _{n=1}^{l} \int _{t^{n-1}}^{t^n} \Vert u^n_{tt}\Vert ^2 d t \right) (\Delta t)^2 + K \sum \limits _{n=1}^{l} \int _{t^{n-1}}^{t^n} \Vert \eta ^n_{t}\Vert ^2 d t\\{} & {} + K (h^r + h^{-1} \Delta t)^2 + 14 \varepsilon \sum \limits _{n=1}^{l} \Vert \partial _t \xi ^n\Vert ^2 \Delta t \\ {}{} & {} + K \sum \limits _{n=1}^{l} (\Vert \xi ^n\Vert ^2 + \Vert \eta ^n\Vert ^2) \Delta t. \end{aligned}$$

Choosing $\varepsilon$ small enough, applying the discrete Gronwall’s lemma and Lemma 3.2, we obtain

$$\begin{aligned} \begin{array}{lll} \alpha _0 \Vert \xi ^l\Vert _1^2 + \sum \limits _{n=1}^{l} \Vert \partial _t \xi ^n\Vert ^2 \Delta t\le & {} K (\Delta t)^2 + K h^{2 (r + 1)} + K (h^r + h^{-1} \Delta t)^2. \end{array} \end{aligned}$$

(3.10)

Let $\Delta t = O(h^{r+1})$. Then, we get

$$\begin{aligned} \begin{array}{lll} \Vert \xi ^l\Vert _1 \le K (\Delta t + h^{r}). \end{array} \end{aligned}$$

(3.11)

Combining the triangle inequality with Lemma 3.2, we get

$$\begin{aligned} \begin{array}{lll} \Vert u^n - u_h^n\Vert _1 \le \Vert \xi ^n\Vert _1 + \Vert \eta ^n\Vert _1 \le K (\Delta t + h^{r}), \end{array} \end{aligned}$$

which is the required bound.

Remark 2

If $2 \le q < \infty$, from (3.4) and similar to the proof of Theorem 3.4 in Xu (1996), we have

$$\begin{aligned} \begin{array}{lll} \Vert u^n - u_h^n \Vert _{1, q} \le K (h^{r} + \Delta t). \end{array} \end{aligned}$$

(3.12)

Furthermore,

$$\begin{aligned} \begin{array}{lll} \Vert u^n - u_h^n \Vert _{0, q} \le K h \Vert u^n - u_h^n \Vert _{1, q} \le K (h^{r + 1} + \Delta t). \end{array} \end{aligned}$$

(3.13)

4 A two-grid discontinuous Galerkin method and its error analysis

Here, we shall present a two-grid algorithm of the discontinuous Galerkin approximation (2.1) to solve the problem (1.1). We use a nonlinear solver on the coarse-grid space with mesh size H and a linear solver on the fine grid with mesh size h ($h \ll H$). The two-grid discontinuous Galerkin method to find $U^n_h \in V_h$ is given in two steps as follows.

Algorithm 1.

Step 1. On the coarse grid, find $u^n_H \in V_H$, such that

$$\begin{aligned} \begin{array}{lll} \left( \frac{u^n_H - u^{n-1}_H}{\Delta t}, v_H\right) + B(u_H^n; u_H^n, v_H) = (f(u_H^n), v_H), \quad v_H \in V_H, \\ u_H^0 = \widetilde{u}_0. \end{array} \end{aligned}$$

Step 2. On the fine grid, find $U^n_h \in V_h$, such that

$$\begin{aligned} \begin{array}{lll} \left( \frac{U^n_h - U^{n-1}_h}{\Delta t}, v_h\right) + B(u_H^n; U_h^n, v_h) = (f(u_H^n) + f'(u_H^n) (U_h^n - u_H^n), v_h), \quad v_h \in V_h, \\ U_h^0 = \widetilde{u}_0. \end{array} \end{aligned}$$

In the above algorithm, we use the discontinuous Galerkin method to solve the nonlinear parabolic problem on a coarse space $V_H$ to get a rough approximation $u^n_H \in V_H$, and use it to linearize the corresponding system on the fine space $V_h$ and solve the resulting linearized problem to get $U^n_h \in V_h$. Using the known solution obtained from Step 1 and the Taylor expansion, the nonlinear problem (2.1) is transformed into a linear problem in Step 2, which is much easier to solve than solving the problem (2.1) on a fine grid directly.

Theorem 4.1

Let $u^n$ and $U_h^n$ be the solution of the problem (1.1) at $t =t^n$ and Algorithm 1, respectively. Let $\Delta t = O(h^{r+1})$. If $U_h^0 = P_h u_0$, then

$$\begin{aligned} \begin{array}{lll} \Vert u^n - U_h^n\Vert \le K (h^{r + 1} +H^{2 (r + 1)} + \Delta t). \end{array} \end{aligned}$$

(4.1)

Proof

For $v \in V_h$, we get the error equation from (1.1) and Algorithm 1

$$\begin{aligned} (u^n_t - \partial _t U_h^n, v) + B(u^n; u^n, v) - B(u^n_H; U_h^n, v) = (f(u^n) - (f(u_H^n) + f'(u_H^n) (U_h^n - u_H^n)), v).\nonumber \\ \end{aligned}$$

(4.2)

Using the notations $\overline{\xi }^n = \widetilde{u}^n - U^n_h$ and $\eta ^n$ used in Sect. 3, choosing $v = \overline{\xi }^{n}$ in the above equation, we get

$$\begin{aligned} (\partial _t \overline{\xi }^n, \overline{\xi }^n) + B_\lambda (u^n_H; \overline{\xi }^n, \overline{\xi }^n)= & {} (\partial _t u^n - u^n_t, \overline{\xi }^n) - (\partial _t \eta ^n, \overline{\xi }^n) + B_\lambda (u^n_H; \widetilde{u}^n, \overline{\xi }^n) \nonumber \\{} & {} - B_\lambda (u^n; \widetilde{u}^n, \overline{\xi }^n) - B_\lambda (u^n; \eta ^n, \overline{\xi }^n) \nonumber \\{} & {} + (f(u^n) - f(u_H^n) - f'(u_H^n) (U_h^n - u_H^n), \overline{\xi }^n) + \lambda (\overline{\xi }^n + \eta ^n, \overline{\xi }^n) \nonumber \\\equiv & {} \sum \limits _{i=1}^{7} R_i. \end{aligned}$$

(4.3)

For the first item on the left hand side of (4.3), we have

$$\begin{aligned} (\partial _t \overline{\xi }^n, \overline{\xi }^n)= & {} \left( \frac{\overline{\xi }^n - \overline{\xi }^{n-1}}{\Delta t}, \overline{\xi }^n \right) \nonumber \\= & {} \frac{1}{2 \Delta t} \left( (\overline{\xi }^n - \overline{\xi }^{n-1}, \overline{\xi }^n + \overline{\xi }^{n-1}) + (\overline{\xi }^n - \overline{\xi }^{n-1}, \overline{\xi }^n - \overline{\xi }^{n-1}) \right) \nonumber \\\ge & {} \frac{1}{2 \Delta t} (\overline{\xi }^n - \overline{\xi }^{n-1}, \overline{\xi }^n + \overline{\xi }^{n-1})\nonumber \\= & {} \frac{1}{2 \Delta t} (\Vert \overline{\xi }^n\Vert ^2 - \Vert \overline{\xi }^{n-1}\Vert ^2). \end{aligned}$$

(4.4)

Next, we will estimate $R_1-R_7$ in (4.3). Applying the Cauchy–Schwarz inequality, we have

$$\begin{aligned} |R_1 \Delta t |\le & {} \Vert \partial _t u^n - u^n_t\Vert \cdot \Vert \overline{\xi }^n\Vert \Delta t \nonumber \\\le & {} K ((\int _{t^{n-1}}^{t^n} \Vert u^n_{tt}\Vert d t)^2 + \Vert \overline{\xi }^n\Vert ^2) \Delta t \nonumber \\\le & {} K \left( \int _{t^{n-1}}^{t^n} \Vert u^n_{tt}\Vert ^2 d t \right) (\Delta t)^2 + K \Vert \overline{\xi }^n\Vert ^2 \Delta t, \nonumber \\ |R_2 \Delta t|\le & {} \Vert \partial _t \eta ^n\Vert \cdot \Vert \overline{\xi }^n\Vert \Delta t \nonumber \\\le & {} K \int _{t^{n-1}}^{t^n} \Vert \eta ^n_{t}\Vert ^2 d t + K \Vert \overline{\xi }^n\Vert ^2 \Delta t. \end{aligned}$$

(4.5)

Similarly to (3.9), using (2.2), we have

$$\begin{aligned} \begin{array}{lll} R_3 + R_4 + R_5\\ \quad = \sum \limits _{E \in \mathcal {T}_{h}}\int _{E} (a(u^n_H)- a(u^n)) \nabla \widetilde{u}^n \cdot \nabla \overline{\xi }^n d x - \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{ (a (u^n_H) - a (u^n)) \nabla \widetilde{u}^n \cdot {\varvec{n}} \} [\overline{\xi }^n] d s\\ \qquad - \sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{(a (u^n_H) - a (u^n)) \nabla \overline{\xi }^n \cdot {\varvec{n}} \} [\widetilde{u}^n] d s\\ \quad \equiv L_1 + L_2 + L_3. \end{array} \end{aligned}$$

For $L_1, L_2$, applying Lemma 3.3, the H$\ddot{o}$lder inequality, the estimate (3.13) and the equality $a(u^n) - a(u^n_H) = a'(\widehat{u}^n) (u^n - u^n_H)$ for some $\widehat{u}^n$ between $u^n$ and $u^n_H$, we have

$$\begin{aligned} |L_1|\le & {} K \Vert \nabla \widetilde{u}^n\Vert _{0,4} \Vert a(u^n_H)- a(u^n)\Vert _{0,4} \Vert \nabla \overline{\xi }^n\Vert \\\le & {} K (\Vert \nabla \widetilde{u}^n\Vert _{0,4}^2 + \Vert a(u^n_H)- a(u^n)\Vert _{0,4}^2) \cdot \Vert \nabla \overline{\xi }^n\Vert \\\le & {} K (\Vert \nabla \widetilde{u}^n\Vert _{\infty }^2 + \Vert u^n_H - u^n\Vert _{0,4}^2) \cdot \Vert \nabla \overline{\xi }^n\Vert \\\le & {} K \Vert u^n_H - u^n\Vert _{0,4}^4 + \varepsilon \Vert \nabla \overline{\xi }^n\Vert ^2 \\\le & {} K (H^{4 (r + 1)} + (\Delta t)^4) + \varepsilon \Vert \overline{\xi }^n\Vert _1^2,\\ |L_2|\le & {} \sum \limits _{\gamma \in \Gamma _{h}} K |\{(u^n - u^n_H) \nabla \widetilde{u}^n \cdot {\varvec{n}}\}|_{\gamma } |[\overline{\xi }^n]|_{\gamma }\\\le & {} K \left( \sum \limits _{\gamma \in \Gamma _{h}} h_{\gamma } \Vert (u^n - u^n_H) \nabla \widetilde{u}^n \cdot {\varvec{n}}\Vert _{\gamma }^2 \right) ^{1/2} \left( \sum \limits _{\gamma \in \Gamma _{h}} h_{\gamma }^{-1} \Vert [\overline{\xi }^n]\Vert _{\gamma }^2 \right) ^{1/2}\\\le & {} K \left( \sum \limits _{\gamma \in \Gamma _{h}} h_{\gamma } \Vert (u^n - u^n_H) \nabla \widetilde{u}^n \cdot {\varvec{n}}\Vert _{\gamma }^2 \right) ^{1/2} \Vert \overline{\xi }^n\Vert _{1}\\\le & {} K \Vert \nabla \widetilde{u}^n\Vert \cdot \Vert u^n - u^n_H\Vert \cdot \Vert \overline{\xi }^n\Vert _{1} \\\le & {} K (\Vert \nabla \widetilde{u}^n\Vert _{0,4}^2 + \Vert u^n - u^n_H\Vert _{0,4}^2) \Vert \overline{\xi }^n\Vert _{1}\\\le & {} K (\Vert \nabla \widetilde{u}^n\Vert _{\infty }^2 + \Vert u^n - u^n_H\Vert _{0,4}^2) \Vert \overline{\xi }^n\Vert _1 \\\le & {} K \Vert u^n - u^n_H\Vert _{0,4}^4 + \varepsilon \Vert \overline{\xi }^n\Vert _1^2\\\le & {} K (H^{4 (r + 1)} + (\Delta t)^4) + \varepsilon \Vert \overline{\xi }^n\Vert _1^2, \end{aligned}$$

For $L_3$, applying the estimate (3.13), we obtain

$$\begin{aligned} |L_3|= & {} |\sum \limits _{\gamma \in \Gamma _{h}} \int _{\gamma } \{(a (u^n_H) - a (u^n)) \nabla \overline{\xi }^n \cdot {\varvec{n}} \} [\eta ^n]|\\\le & {} \sum \limits _{\gamma \in \Gamma _{h}} |\{(a (u^n_H) - a (u^n)) \nabla \overline{\xi }^n \cdot {\varvec{n}} \}|_{\gamma } |[\eta ^n]|_{\gamma }\\\le & {} K \sum \limits _{\gamma \in \Gamma _{h}} |\nabla \overline{\xi }^n \cdot {\varvec{n}}|_{\gamma } |\{u^n_H - u^n\}|_{\gamma } \cdot |[\eta ^n]|_{\gamma }\\\le & {} K h^{-1} |\nabla \overline{\xi }^n| |u^n_H - u^n| \cdot |\eta ^n|\\\le & {} K (h^{-1} \Vert \eta ^n\Vert _{0,4} \Vert u^n - u^n_H\Vert _{0,4}) \Vert \nabla \overline{\xi }^n\Vert \\\le & {} K (\Vert \eta ^n\Vert _{1}^2 + \Vert u^n - u^n_H\Vert _{0,4}^2) \Vert \nabla \overline{\xi }^n\Vert \\\le & {} K \Vert u^n - u^n_H\Vert _{0,4}^4 + \varepsilon \Vert \overline{\xi }^n\Vert _1^2 + K \Vert \eta ^n\Vert _{1}^4\\\le & {} K (H^{4 (r + 1)} + (\Delta t)^4) + \varepsilon \Vert \overline{\xi }^n\Vert _1^2 + + K \Vert \eta ^n\Vert _{1}^4, \end{aligned}$$

where the inverse inequality is used.

Note that a Taylor expansion yields

$$\begin{aligned}{} & {} f(u^n) - f(u_H^n) - f'(u_H^n) (U_h^n - u_H^n) \\{} & {} \quad = f(u_H^n) + f'(u_H^n) (u^n - u_H^n) + \frac{1}{2} f''(\widehat{u}^n) (u^n - u_H^n)^2 - f(u_H^n) - f'(u_H^n) (U_h^n - u_H^n)\\{} & {} \quad = f'(u_H^n) (u^n - U_h^n) + \frac{1}{2} f''(\widehat{u}^n) (u^n - u_H^n)^2\\{} & {} \quad = f'(u_H^n) (\overline{\xi }^n + \eta ^n) + \frac{1}{2} f''(\widehat{u}^n) (u^n - u_H^n)^2, \end{aligned}$$

for some $\widehat{u}^n$ between $u^n$ and $u_H^n$. Then, by virtue of the boundness of $f''(\cdot )$, we get

$$\begin{aligned} R_6= & {} (f(u^n) - f(u_H^n) - f'(u_H^n) (U_h^n - u_H^n), \overline{\xi }^n)\\\le & {} K (\Vert \overline{\xi }^n\Vert ^2 + \Vert \eta ^n\Vert ^2) + K \Vert (u^n - u_H^n)^2\Vert ^2 + K \Vert \overline{\xi }^n\Vert ^2\\\le & {} K (\Vert \overline{\xi }^n\Vert ^2 + \Vert \eta ^n\Vert ^2) + K (H^{4 (r + 1)} + (\Delta t)^4) + K \Vert \overline{\xi }^n\Vert ^2. \end{aligned}$$

And,

$$\begin{aligned} R_7 \le K (\Vert \overline{\xi }^n\Vert ^2 + \Vert \eta ^n\Vert ^2). \end{aligned}$$

Multiplying by $2 \Delta t$ and summing Eq. (4.3) from $n=1$ to $n=l \quad (1 \le l \le N)$, collecting all the above estimates and using Lemma 2.3, we obtain

$$\begin{aligned}{} & {} \Vert \overline{\xi }^l\Vert ^2 + 2 \alpha _0 \sum \limits _{n=1}^{l} \Vert \overline{\xi }^n\Vert _1^2 \Delta t \nonumber \\{} & {} \quad \le K \left( \sum \limits _{n=1}^{l} \int _{t^{n-1}}^{t^n} \Vert u^n_{tt}\Vert ^2 d t \right) (\Delta t)^2 + K \sum \limits _{n=1}^{l} \int _{t^{n-1}}^{t^n} \Vert \eta ^n_{t}\Vert ^2 d t \nonumber \\{} & {} \qquad + K (H^{4 r + 4} + (\Delta t)^4) + 6 \varepsilon \sum \limits _{n=1}^{l} \Vert \overline{\xi }^n\Vert _1^2 \Delta t \nonumber \\{} & {} \qquad + K \sum \limits _{n=1}^{l} (\Vert \eta ^n\Vert ^2 + \Vert \eta ^n\Vert _1^4 + \Vert \overline{\xi }^n\Vert ^2) \Delta t. \end{aligned}$$

(4.6)

For $r \ge 1$, choosing $\varepsilon$ small enough and applying Gronwall’s lemma, we get

$$\begin{aligned} \begin{array}{lll} \Vert \overline{\xi }^l\Vert ^2 + 2 \alpha _0 \sum \limits _{n=1}^{l} \Vert \overline{\xi }^n\Vert _1^2 \Delta t \le K (h^{2 (r + 1)} + H^{4( r + 1)} + (\Delta t)^2). \end{array} \end{aligned}$$

(4.7)

It is obvious that

$$\begin{aligned} \Vert \overline{\xi }^l\Vert \le K (h^{r + 1} + H^{2( r + 1)} + \Delta t). \end{aligned}$$

(4.8)

Using the triangle inequality with Lemma 3.2, we obtain (4.1), which completes the proof.

Remark 3

Theorem 4.1 indicates that the optimal convergence rate in the full-discrete two-grid discontinuous Galerkin approximation using the $r \ge 1$th order discontinuous finite element space can be achieved if $h = O(H^{2})$.

5 Numerical experiments

Here, we shall solve problem (1.1) using the proposed two-grid discontinuous Galerkin method in Matlab.

For problem (1.1), we take $a(u) = e^u$, $f(u)= u^3 + g(x, t)$, where g(x, t) is decided so that the exact solution of (1.1) is $u(x, t) = t^2 (x_1^2 (x_1 - 1)^2 + x_2^2 (x_2 - 1)^2 )$ for $x = (x_1, x_2) \in \Omega = (0,1) \times (0,1)$.

The simulation time is $T = 0.1$ and we discretize the time variable with a time step $\Delta t = 1.0e{-}4$. To solve this problem, we use Algorithm 1 with $\sigma _{\gamma } = 10.0$ and with the piecewise linear and quadratic discontinuous finite element space ($r=1$ and $r=2$).

The domain $\Omega$ is uniformly divided by the triangulation of mesh size H and h, where H and h are the space step of the coarse grid and the fine grid, respectively. On the coarse grid, nonlinear algebraic equations are solved by the Newton method to get the next iterative solution from the current solution. In each time interval $[t_{m-1}, t_m]$, we use the stopping criterion $\sum \nolimits _{i=1}^{(N+1)^2} (U_{h, i}^m - U_{h, i}^{m-1})^2 \le 10^{-8}$, where N is the number of nodes in each orientation and m is the step of the iteration. We give the errors at $t = T$ and the CPU time of the two-grid discontinuous Galerkin method and the standard discontinuous Galerkin (SDG) method in Tables 1 and 2. The convergence results of Theorem 4.1 are verified. The convergence rate is denoted by $rate\,=\,\log _{2}(\frac{\delta _{N}}{\delta _{2N}})$, where $\delta _{N}$ is the error of a fixed N.

Table 1 $L_2$ errors and CPU time of the standard discontinuous Galerkin method (SDG) and the two-grid discontinuous Galerkin method (two-grid DG) with $r=1$

Table 2 errors and CPU time of the standard discontinuous Galerkin method (SDG) and the two-grid discontinuous Galerkin method (two-grid DG) with $r=2$

It can be seen that the order of convergence of the two-grid discontinuous Galerkin method with $r=1$ and $r=2$ are about 2 and 3, respectively. The two-grid discontinuous Galerkin method spends less time than the standard discontinuous Galerkin method to achieve the same accuracy from the above data. Our proposed two-grid discontinuous Galerkin algorithm is effective. The numerical analysis coincides with the theoretical analysis.

6 Conclusions

In the paper, we have presented the error estimates of a full-discrete discontinuous Galerkin method and its two-grid algorithm for the nonlinear parabolic problem (1.1). With the two-grid method, a large amount of computational cost was saved because the nonlinear system is only solved on a coarse grid with mesh size H and an easy linear problem can be solved on a fine grid with mesh size h. The analysis shows that the optimal convergence rate in the full-discrete two-grid discontinuous Galerkin approximation using the $r \ge 1$ order discontinuous finite element space can be achieved by employing $h = O(H^{2})$.