Unconditional superconvergence analysis of an energy-stable finite element scheme for nonlinear Benjamin–Bona–Mahony–Burgers equation

Abstract

In this paper, an energy-stable Crank–Nicolson fully discrete finite element scheme is proposed for the Benjamin–Bona–Mahony–Burgers equation. Firstly, the stability of energy is proved, which leads to the boundedness of the finite element solution in $H^{1}$-norm. Secondly, combining with the above boundedness and the special property of bilinear element, the unconditional superclose and superconvergence results are derived. Finally, numerical examples are provided to illustrate the validity and efficiency of our theoretical analysis and method.

1 Introduction

The nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation is often used to describe the propagation of small amplitude long waves in a nonlinear dispersive medium with dissipative effect, which is considered as the following second-order partial differential equation [1]:

$$\begin{aligned} \textstyle\begin{cases} u_{t}-\alpha \Delta u_{t}-\beta \Delta u=\nabla \cdot \vec{f}(u), & (X,t)\in \Omega \times (0,T], \\ u(X,t)=0, &(X,t)\in \partial \Omega \times (0,T], \\ { u(X,0)}=u_{0}(X), & X\in \Omega , \end{cases}\displaystyle \end{aligned}$$

(1)

where $\Omega \subset R^{2}$ is an open bounded convex polygonal domain with boundary ∂Ω, $\alpha >0$, $\beta >0$, $0< T<\infty$ are given constants, $X=(x,y)$, $\vec{f}(u)=-(\frac{1}{2}u^{2}+u,\frac{1}{2}u^{2}+u)$, Δ and ∇⋅ denote the two-dimensional Laplace and divergence operators, respectively, $u_{0}(X)$ is a given smooth function. It is remarkable that when $\alpha =0$, $\beta >0$, (1) is called Burgers’ equation, when $\alpha >0$, $\beta =0$, (1) is called Benjamin–Bona–Mahony (BBM) equation. Various analytical and computational methods have been proposed to solve Burgers’ and BBM equations, readers with more interests may refer to [2–7] and the references listed.

Due to the nonlinearity of BBMB equation, it is very difficult to find out the true solution. Thus, a lot of numerical methods have been considered, such as the finite difference methods [8–11], collocation method [12], meshless method [13, 14], finite element method (FEM) [15–18], and so on. For the FEM, Kadri [15] proposed semi-discrete and two kinds of fully discrete Galerkin schemes, studied the $L^{\infty}$-norm error estimates; Kundu [16] established the convergence of unsteady solution to steady state solution; Karakoc [17] obtained the convergence analysis by use of a cubic B-spline FEM; Gao [18] discussed the local discontinuous Galerkin FEM and derived an optimal error estimate. However, [15–18] only focus on the convergence for the one-dimensional (1D) BBMB equation, there are few works for the 2D case till now.

It is well known that the superconvergence analysis is an important approach to improve the precision of FE solution. More precisely, based on the so-called integral identity technique, the order of error in $H^{1}$-norm between FE approximation $u_{h}$ and the interpolation of the exact solution $I_{h}u$ is much better than that of u and $I_{h}u$; this fascinating characteristic is called superclose. The global superconvergence will then be investigated by adding a postprocessing without changing the existing FE program. Meanwhile, superconvergence is critical in practical engineering numerical calculation and has always been a research hotspot. To find out more applications, readers may refer to [19–23]. As far as our knowledge is concerned, research on superconvergence for 2D BBMB equation is not yet to be found.

In this work, as a first attempt, we develop an energy-stable conforming FE scheme for problem (1) and study the superclose and superconvergence error estimates. The outline is organized as follows: in Sect. 2, the FE space and the Crank–Nicolson (C-N) fully discrete scheme are provided, then the stability of energy and the boundedness of the numerical solution in $H^{1}$-norm are proved; in Sect. 3, the unconditional superclose and superconvergence results are derived without the restriction of the ratio between mesh size parameter h and time step Δt; in the last section, three numerical examples are given to verify the theoretical analysis.

2 The FE space and energy-stable scheme

Assume that $W^{k,p}(\Omega )$ is the standard Sobolev space with the norm $\|\cdot \|_{W^{k,p}(\Omega )}$, $H^{k}(\Omega )= W^{k,2}(\Omega )$, $H^{0}(\Omega )= L^{2}(\Omega )$ with the norm $\|\cdot \|_{k}$ and $\|\cdot \|_{0}$, the inner-product in $L^{2}(\Omega )$ is defined by $( \cdot , \cdot )$.

Denote $T_{h}$ to be a regular rectangular subdivision of Ω. For $K \in T_{h}$, $h_{K}=\operatorname{diam}K$, $h=\max_{K\in{T}_{h}}h_{K}$. The bilinear element space $V_{h}$ is defined by

$$\begin{aligned} V_{h}=\bigl\{ v_{h}\in H^{1}_{0}(\Omega ) : v_{h}|_{K}\in span\{1,x,y,xy\}, v_{h}|_{ \partial \Omega}=0, \forall K \in T_{h}\bigr\} . \end{aligned}$$

The associated interpolation operator is defined as $I_{h}:v\in V=H^{1}_{0}( \Omega )\rightarrow I_{h}v \in V_{h}$.

The variational form of (1) is to find $u\in V$ such that

$$\begin{aligned} \textstyle\begin{cases} (u_{t},v)+\alpha (\nabla u_{t},\nabla v)+\beta (\nabla u,\nabla v)=-( \vec{f}(u),\nabla v), \quad \forall v\in V, \\ u(X,0)=u_{0}(X). \end{cases}\displaystyle \end{aligned}$$

(2)

Let $\{t_{n}|t_{n}=n\Delta t; n=0,1,2,\ldots , N\}$ be a uniform partition of $[0,T]$ with $\Delta t=T/N$. For a given continuous function u on $[0,T]$, we define that $u^{n}=u(X,t_{n})$, $\bar{\partial}_{t}u^{n}= \frac{u^{n}-u^{n-1}}{\Delta t}$, $t_{n-\frac{1}{2}}=(n-\frac{1}{2})\Delta t$, ${u}^{n-\frac{1}{2}}=\frac{u^{n}+u^{n-1}}{2}$. The C-N fully discrete scheme for (2) is to find $U_{h}^{n}\in V_{h}$ such that

$$\begin{aligned} \textstyle\begin{cases} (\bar{\partial}_{t}U^{n}_{h},v_{h})+\alpha (\nabla \bar{\partial}_{t}U^{n}_{h}, \nabla v_{h})+\beta (\nabla U^{n-\frac{1}{2}}_{h},\nabla v_{h})=-( \vec{f}(U^{n-\frac{1}{2}}_{h}),\nabla v_{h}), \quad \forall v_{h} \in V_{h}, \\ U^{n}_{h}(X,0)=I_{h}u_{0}(X). \end{cases}\displaystyle \end{aligned}$$

(3)

First of all, we achieve the following special properties of bilinear element from [19, 24].

Lemma 2.1

For all $v_{h}\in V_{h}$, there hold

$$\begin{aligned}& \Vert u-I_{h}u \Vert _{0,p}\leq Ch^{2} \Vert u \Vert _{2,p}, \quad u\in W^{2,p}( \Omega ), \end{aligned}$$

(4)

$$\begin{aligned}& \bigl(\nabla (u-I_{h}u),\nabla v_{h}\bigr)\leq Ch^{2} \Vert u \Vert _{3} \Vert v_{h} \Vert _{1}, \quad u \in H^{3}(\Omega ). \end{aligned}$$

(5)

Here and later, we denote by C a generic positive constant which is independent of h and Δt and may stand for different values at different places.

Then the energy stability of (3) and the boundedness of $\|U_{h}^{n}\|_{1}$ are proved as follows.

Theorem 2.1

Let $E^{n}=\|U_{h}^{n}\|^{2}_{0}+\alpha |U_{h}^{n}|^{2}_{1}$ ($n=0, 1, \ldots , N$) be the discrete energy, $U_{h}^{n}$ is the solution of (3), then there holds

$$\begin{aligned} E^{n}\leq E^{n-1} \quad (n=1,2, \ldots , N). \end{aligned}$$

(6)

Furthermore, we obtain

$$\begin{aligned} \bigl\Vert U_{h}^{n} \bigr\Vert _{1}\leq M_{1}, \end{aligned}$$

(7)

where $M_{1}=\sqrt{\frac{\max \{1,\alpha \}}{\min \{1,\alpha \}}}\|U^{0}_{h} \|_{1}$ is a positive constant.

Proof

Taking $v_{h}=U_{h}^{n-\frac{1}{2}}$ in (3), we can get

$$\begin{aligned} \bigl(\bar{\partial}_{t} U_{h}^{n}, U_{h}^{n-\frac{1}{2}}\bigr)+\alpha \bigl(\nabla \bar{ \partial}_{t} U_{h}^{n}, \nabla U_{h}^{n-\frac{1}{2}}\bigr)+\beta \bigl( \nabla U_{h}^{n-\frac{1}{2}}, \nabla U_{h}^{n-\frac{1}{2}}\bigr)=-\bigl(\vec{f}\bigl(U^{n- \frac{1}{2}}_{h} \bigr),\nabla U_{h}^{n-\frac{1}{2}}\bigr). \end{aligned}$$

(8)

Firstly, the left-hand side of (8) can be rewritten as

$$\begin{aligned} \frac{1}{2\Delta t}\bigl( \bigl\Vert U_{h}^{n} \bigr\Vert ^{2}_{0}- \bigl\Vert U_{h}^{n-1} \bigr\Vert ^{2}_{0}\bigr)+ \frac{\alpha}{2\Delta t}\bigl( \bigl\vert U_{h}^{n} \bigr\vert ^{2}_{1}- \bigl\vert U_{h}^{n-1} \bigr\vert ^{2}_{1} \bigr)+ \beta \bigl\Vert \nabla U_{h}^{n-\frac{1}{2}} \bigr\Vert ^{2}_{0}. \end{aligned}$$

(9)

Secondly, the right-hand side can be split as

$$\begin{aligned} -\bigl(\vec{f}\bigl(U^{n-\frac{1}{2}}_{h}\bigr),\nabla U_{h}^{n-\frac{1}{2}}\bigr)& = \int _{ \Omega} \biggl(\frac{1}{2}\bigl(U^{n-\frac{1}{2}}_{h} \bigr)^{2}+U^{n-\frac{1}{2}}_{h}\biggr)\bigl[ \bigl(U^{n- \frac{1}{2}}_{h}\bigr)_{x}+\bigl(U^{n-\frac{1}{2}}_{h} \bigr)_{y}\bigr]\,dx \,dy \\ & = \int _{\Omega} \frac{1}{2}\bigl(U^{n-\frac{1}{2}}_{h} \bigr)^{2}\bigl[\bigl(U^{n- \frac{1}{2}}_{h} \bigr)_{x}+\bigl(U^{n-\frac{1}{2}}_{h}\bigr)_{y} \bigr]\,dx \,dy \\ &\quad{}+ \int _{\Omega} U^{n-\frac{1}{2}}_{h}\bigl[ \bigl(U^{n-\frac{1}{2}}_{h}\bigr)_{x}+\bigl(U^{n- \frac{1}{2}}_{h} \bigr)_{y}\bigr]\,dx \,dy. \end{aligned}$$

(10)

By using the Green formula and noting that $U^{n-\frac{1}{2}}_{h}|_{\partial \Omega}=0$, we obtain

$$\begin{aligned} & \int _{\Omega} \bigl(U^{n-\frac{1}{2}}_{h} \bigr)^{2}\bigl[\bigl(U^{n-\frac{1}{2}}_{h} \bigr)_{x}+\bigl(U^{n- \frac{1}{2}}_{h}\bigr)_{y} \bigr]\,dx \,dy \\ &\quad = - \int _{\Omega}\bigl(U^{n-\frac{1}{2}}_{h}\bigr) \bigl[2U^{n- \frac{1}{2}}_{h}\bigl(U^{n-\frac{1}{2}}_{h} \bigr)_{x}+2U^{n-\frac{1}{2}}_{h}\bigl(U^{n- \frac{1}{2}}_{h} \bigr)_{y}\bigr]\,dx \,dy \\ &\qquad{}+ \int _{\partial \Omega} U^{n-\frac{1}{2}}_{h}\bigl(U^{n-\frac{1}{2}}_{h} \bigr)^{2} \cdot \vec{n}\,ds \\ &\quad = - \int _{\Omega}2\bigl(U^{n-\frac{1}{2}}_{h} \bigr)^{2}\bigl[\bigl(U^{n-\frac{1}{2}}_{h} \bigr)_{x}+\bigl(U^{n- \frac{1}{2}}_{h}\bigr)_{y} \bigr]\,dx \,dy, \end{aligned}$$

where n⃗ is the outer normal vector, so we arrive at

$$\begin{aligned} \int _{\Omega} \bigl(U^{n-\frac{1}{2}}_{h} \bigr)^{2}\bigl[\bigl(U^{n-\frac{1}{2}}_{h} \bigr)_{x}+\bigl(U^{n- \frac{1}{2}}_{h}\bigr)_{y} \bigr]\,dx \,dy=0. \end{aligned}$$

(11)

Similarly, we have

$$\begin{aligned} \int _{\Omega} U^{n-\frac{1}{2}}_{h}\bigl[ \bigl(U^{n-\frac{1}{2}}_{h}\bigr)_{x}+\bigl(U^{n- \frac{1}{2}}_{h} \bigr)_{y}\bigr]\,dx \,dy=0, \end{aligned}$$

(12)

substituting (11) and (12) into (10), we obtain

$$\begin{aligned} \bigl(\vec{f}\bigl(U^{n-\frac{1}{2}}_{h}\bigr),\nabla U_{h}^{n-\frac{1}{2}}\bigr)=0. \end{aligned}$$

(13)

From (8), (9), and (13), there holds

$$\begin{aligned} \frac{1}{2\Delta t}\bigl[\bigl( \bigl\Vert U_{h}^{n} \bigr\Vert ^{2}_{0}- \bigl\Vert U_{h}^{n-1} \bigr\Vert ^{2}_{0}\bigr)+ \alpha \bigl( \bigl\vert U_{h}^{n} \bigr\vert ^{2}_{1}- \bigl\vert U_{h}^{n-1} \bigr\vert ^{2}_{1} \bigr)\bigr]+\beta \bigl\Vert \nabla U_{h}^{n- \frac{1}{2}} \bigr\Vert ^{2}_{0}=0, \end{aligned}$$

(14)

therefore, we have

$$\begin{aligned} \bigl\Vert U_{h}^{n} \bigr\Vert ^{2}_{0}+ \alpha \bigl\vert U_{h}^{n} \bigr\vert ^{2}_{1}+2\Delta t\beta \bigl\Vert \nabla U_{h}^{n-\frac{1}{2}} \bigr\Vert ^{2}_{0}= \bigl\Vert U_{h}^{n-1} \bigr\Vert ^{2}_{0}+ \alpha \bigl\vert U_{h}^{n-1} \bigr\vert ^{2}_{1}, \end{aligned}$$

which implies $E^{n}\leq E^{n-1}$, (6) is obtained.

Next we start to demonstrate (7). Multiplying $2\Delta t$ on both sides of (14), replacing n with i, and summing for i from 1 to n, we have

$$\begin{aligned} \bigl\Vert U_{h}^{n} \bigr\Vert ^{2}_{0}+ \alpha \bigl\vert U_{h}^{n} \bigr\vert ^{2}_{1}+2\Delta t\beta \sum^{n}_{i=1} \bigl\Vert \nabla U_{h}^{i-\frac{1}{2}} \bigr\Vert ^{2}_{0}= \bigl\Vert U_{h}^{0} \bigr\Vert ^{2}_{0}+\alpha \bigl\vert U_{h}^{0} \bigr\vert ^{2}_{1}. \end{aligned}$$

(15)

From (15) and the triangular inequality, we have

$$\begin{aligned} \min \{1,\alpha \} \bigl\Vert U_{h}^{n} \bigr\Vert ^{2}_{1}\leq \max \{1,\alpha \} \bigl\Vert U_{h}^{0} \bigr\Vert _{1}^{2}, \end{aligned}$$

which ends the proof. □

3 Superclose and superconvergence analysis

We first demonstrate the following unconditional superclose result.

Theorem 3.1

Let $u^{n}$ and $U_{h}^{n}$ be solutions of (2) and (3), respectively. Assume that $u\in L^{\infty}(0,T;H^{3}(\Omega ))$, $u_{t}\in L^{2}(0,T;H^{3}(\Omega ))$, $u_{tt},u_{ttt}\in L^{2}(0,T;H^{1}(\Omega ))$, there holds

$$\begin{aligned} \bigl\Vert I_{h}u^{n}-U_{h}^{n} \bigr\Vert _{1}\leq C\bigl(h^{2}+(\Delta t)^{2} \bigr), \end{aligned}$$

(16)

where $\Delta t>0$ is small enough so that $1-C\Delta t>0$.

Proof

Let $u^{n}-U^{n}_{h}=(u^{n}-I_{h}u^{n})+(I_{h}u^{n}-U_{h}^{n}):= \xi ^{n}+ \eta ^{n}$, then the error equation can be derived by (2) and (3):

$$\begin{aligned} &\bigl(\bar{\partial}_{t}\eta ^{n},v_{h}\bigr)+ \alpha \bigl(\nabla \bar{\partial}_{t} \eta ^{n},\nabla v_{h}\bigr)+\beta \bigl(\nabla \eta ^{n-\frac{1}{2}},\nabla v_{h}\bigr) \\ &\quad=-\bigl(\bar{\partial}_{t}\xi ^{n},v_{h}\bigr)- \alpha \bigl(\nabla \bar{\partial}_{t} \xi ^{n},\nabla v_{h}\bigr)-\beta \bigl(\nabla \xi ^{n-\frac{1}{2}},\nabla v_{h}\bigr) \\ &\quad\quad{}+\bigl(\vec{f}\bigl(U^{n-\frac{1}{2}}_{h}\bigr)-\vec{f} \bigl(u(t_{n-\frac{1}{2}})\bigr), \nabla v_{h}\bigr)- \bigl(R_{1}^{n},v_{h}\bigr) \\ &\quad\quad{}-\alpha \bigl(\nabla R_{1}^{n},\nabla v_{h} \bigr)-\beta \bigl(\nabla R_{2}^{n}, \nabla v_{h} \bigr), \end{aligned}$$

(17)

where $R^{n}_{1}=u_{t}(t_{n-\frac{1}{2}})-\bar{\partial}_{t}u^{n}$, $R_{2}^{n}=u(t_{n- \frac{1}{2}})-u^{n-\frac{1}{2}}$.

Taking $v_{h}=\eta ^{n-\frac{1}{2}}$ in (17), there holds

$$\begin{aligned} &\bigl(\bar{\partial}_{t}\eta ^{n},\eta ^{n-\frac{1}{2}} \bigr)+\alpha \bigl(\nabla \bar{\partial}_{t}\eta ^{n},\nabla \eta ^{n-\frac{1}{2}}\bigr)+\beta \bigl( \nabla \eta ^{n-\frac{1}{2}},\nabla \eta ^{n-\frac{1}{2}}\bigr) \\ &\quad=-\bigl(\bar{\partial}_{t}\xi ^{n},\eta ^{n-\frac{1}{2}} \bigr)-\alpha \bigl(\nabla \bar{\partial}_{t}\xi ^{n},\nabla \eta ^{n-\frac{1}{2}}\bigr) \\ &\quad\quad{}-\beta \bigl(\nabla \xi ^{n-\frac{1}{2}},\nabla \eta ^{n-\frac{1}{2}}\bigr) + \bigl( \vec{f}\bigl(U^{n-\frac{1}{2}}_{h}\bigr)-\vec{f} \bigl(u(t_{n-\frac{1}{2}})\bigr),\nabla \eta ^{n-\frac{1}{2}}\bigr) \\ &\quad\quad{}-\bigl(R_{1}^{n},\eta ^{n-\frac{1}{2}}\bigr)-\alpha \bigl( \nabla R_{1}^{n},\nabla \eta ^{n-\frac{1}{2}}\bigr) -\beta \bigl(\nabla R_{2}^{n},\nabla \eta ^{n- \frac{1}{2}}\bigr) . \end{aligned}$$

(18)

The left-hand side of (18) can be rewritten as

$$\begin{aligned} \frac{1}{2\Delta t}\bigl[\bigl( \bigl\Vert \eta ^{n} \bigr\Vert ^{2}_{0}- \bigl\Vert \eta ^{n-1} \bigr\Vert ^{2}_{0}\bigr)+ \alpha \bigl( \bigl\Vert \nabla \eta ^{n} \bigr\Vert ^{2}_{0}- \bigl\Vert \nabla \eta ^{n-1} \bigr\Vert ^{2}_{0}\bigr)\bigr] + \beta \bigl\Vert \nabla \eta ^{n-\frac{1}{2}} \bigr\Vert ^{2}_{0}. \end{aligned}$$

(19)

Now we estimate the right-hand side: By virtue of Lemma 2.1, we arrive at

$$\begin{aligned} &\bigl(\bar{\partial}_{t}\xi ^{n},\eta ^{n-\frac{1}{2}} \bigr)+\alpha \bigl(\nabla \bar{\partial}_{t}\xi ^{n},\nabla \eta ^{n-\frac{1}{2}}\bigr)+\beta \bigl( \nabla \xi ^{n-\frac{1}{2}},\nabla \eta ^{n-\frac{1}{2}}\bigr) \\ &\quad \leq Ch^{4}\biggl(\frac{1}{\Delta t} \int ^{t_{n}}_{t_{n-1}} \Vert u_{t} \Vert ^{2}_{3}\,d \tau + \bigl\Vert u^{n-\frac{1}{2}} \bigr\Vert ^{2}_{3}\biggr)+\frac{1}{2} \bigl\Vert \nabla \eta ^{n- \frac{1}{2}} \bigr\Vert ^{2}_{0}. \end{aligned}$$

(20)

Using the Taylor expansion, the truncation error can be estimated as

$$\begin{aligned} &\bigl( R_{1}^{n},\eta ^{n-\frac{1}{2}}\bigr)+\alpha \bigl( \nabla R_{1}^{n},\nabla \eta ^{n-\frac{1}{2}}\bigr)+\beta \bigl(\nabla R_{2}^{n},\nabla \eta ^{n- \frac{1}{2}}\bigr) \\ &\quad\leq C\bigl( \bigl\Vert R_{1}^{n} \bigr\Vert ^{2}_{1}+ \bigl\Vert R_{2}^{n} \bigr\Vert ^{2}_{1}\bigr)+\frac{1}{2} \bigl\Vert \nabla \eta ^{n-\frac{1}{2}} \bigr\Vert ^{2}_{0} \\ &\quad\leq C(\Delta t)^{3} \int ^{t_{n}}_{t_{n-1}}\bigl( \Vert u_{ttt} \Vert ^{2}_{1}+ \Vert u_{tt} \Vert ^{2}_{1}\bigr)\,d\tau +\frac{1}{2} \bigl\Vert \nabla \eta ^{n-\frac{1}{2}} \bigr\Vert ^{2}_{0}, \end{aligned}$$

(21)

the nonlinear term can be written as

$$\begin{aligned} & \bigl\vert \bigl(\vec{f}\bigl(U^{n-\frac{1}{2}}_{h}\bigr)-\vec{f} \bigl(u(t_{n-\frac{1}{2}})\bigr), \nabla \eta ^{n-\frac{1}{2}}\bigr) \bigr\vert \\ &\quad = \biggl\vert \biggl(\frac{1}{2}\bigl(\bigl(u(t_{n-\frac{1}{2}}) \bigr)^{2}-\bigl(U^{n-\frac{1}{2}}_{h}\bigr)^{2} \bigr), \nabla \eta ^{n-\frac{1}{2}}\biggr)+\bigl(u(t_{n-\frac{1}{2}})-U^{n-\frac{1}{2}}_{h}, \nabla \eta ^{n-\frac{1}{2}}\bigr) \biggr\vert \\ &\quad = \biggl\vert \biggl(\frac{1}{2}\bigl(\bigl(u(t_{n-\frac{1}{2}}) \bigr)^{2}-\bigl(u^{n-\frac{1}{2}}\bigr)^{2}+ \bigl(u^{n- \frac{1}{2}}\bigr)^{2}-\bigl(U^{n-\frac{1}{2}}_{h} \bigr)^{2}\bigr), \nabla \eta ^{n- \frac{1}{2}}\biggr) \\ &\quad\quad{}+\bigl(u(t_{n-\frac{1}{2}})-u^{n-\frac{1}{2}}+u^{n-\frac{1}{2}}-U^{n- \frac{1}{2}}_{h}, \nabla \eta ^{n-\frac{1}{2}}\bigr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{2}\bigl(u(t_{n-\frac{1}{2}})^{2}- \bigl(u^{n-\frac{1}{2}}\bigr)^{2}, \nabla \eta ^{n-\frac{1}{2}}\bigr)+ \frac{1}{2}\bigl(\bigl(u^{n-\frac{1}{2}}\bigr)^{2}- \bigl(U^{n- \frac{1}{2}}_{h}\bigr)^{2}, \nabla \eta ^{n-\frac{1}{2}}\bigr) \\ &\quad\quad{}+\bigl(u(t_{n-\frac{1}{2}})-u^{n-\frac{1}{2}}+u^{n-\frac{1}{2}}-U^{n- \frac{1}{2}}_{h}, \nabla \eta ^{n-\frac{1}{2}}\bigr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{2}\bigl(R^{n}_{2} \bigl(u(t_{n-\frac{1}{2}})+u^{n-\frac{1}{2}}\bigr), \nabla \eta ^{n-\frac{1}{2}} \bigr)+\frac{1}{2}\bigl(\bigl(\xi ^{n-\frac{1}{2}}+\eta ^{n- \frac{1}{2}}\bigr) \bigl(u^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}_{h}\bigr), \nabla \eta ^{n- \frac{1}{2}}\bigr) \\ &\quad\quad{}+\bigl(R^{n}_{2}, \nabla \eta ^{n-\frac{1}{2}}\bigr)+\bigl( \xi ^{n-\frac{1}{2}}+ \eta ^{n-\frac{1}{2}},\nabla \eta ^{n-\frac{1}{2}}\bigr) \biggr\vert \\ &\quad = \Biggl\vert \sum^{4}_{i=1}A_{i} \Biggr\vert . \end{aligned}$$

(22)

By the estimation of truncation error, there holds

$$\begin{aligned} A_{1}+A_{3}&\leq C \bigl\Vert R^{n}_{2} \bigr\Vert _{0} \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert _{1}\bigl( \bigl\Vert u(t_{n- \frac{1}{2}})+u^{n-\frac{1}{2}} \bigr\Vert _{0,\infty}+1\bigr) \\ &\leq C \bigl\Vert R^{n}_{2} \bigr\Vert _{0} \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert _{1} \\ &\leq C(\Delta t)^{3} \int ^{t_{n}}_{t_{n-1}} \Vert u_{tt} \Vert ^{2}_{0}\,d\tau +C \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert ^{2}_{1}. \end{aligned}$$

(23)

From the Sobolev imbedding theorem, (4) and (7), we have

$$\begin{aligned} A_{2}& \leq C \bigl\Vert \xi ^{n-\frac{1}{2}}+\eta ^{n-\frac{1}{2}} \bigr\Vert _{0,4} \bigl\Vert u^{n- \frac{1}{2}}+U^{n-\frac{1}{2}}_{h} \bigr\Vert _{0,4} \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert _{1} \\ & \leq C \bigl\Vert \xi ^{n-\frac{1}{2}}+\eta ^{n-\frac{1}{2}} \bigr\Vert _{0,4}\bigl( \bigl\Vert u^{n- \frac{1}{2}} \bigr\Vert _{0,4}+ \bigl\Vert U^{n-\frac{1}{2}}_{h} \bigr\Vert _{0,4}\bigr) \bigl\Vert \eta ^{n- \frac{1}{2}} \bigr\Vert _{1} \\ & \leq C \bigl\Vert \xi ^{n-\frac{1}{2}}+\eta ^{n-\frac{1}{2}} \bigr\Vert _{0,4}\bigl( \bigl\Vert u^{n- \frac{1}{2}} \bigr\Vert _{0,\infty}+ \bigl\Vert U^{n-\frac{1}{2}}_{h} \bigr\Vert _{1}\bigr) \bigl\Vert \eta ^{n- \frac{1}{2}} \bigr\Vert _{1} \\ & \leq C \bigl\Vert \xi ^{n-\frac{1}{2}}+\eta ^{n-\frac{1}{2}} \bigr\Vert _{0,4} \bigl\Vert \eta ^{n- \frac{1}{2}} \bigr\Vert _{1} \\ & \leq Ch^{4} \bigl\Vert u^{n-\frac{1}{2}} \bigr\Vert ^{2}_{2,4}+C \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert ^{2}_{1} \\ & \leq Ch^{4} \bigl\Vert u^{n-\frac{1}{2}} \bigr\Vert ^{2}_{3}+C \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert ^{2}_{1} \end{aligned}$$

(24)

and

$$\begin{aligned} A_{4}\leq C\bigl( \bigl\Vert \xi ^{n-\frac{1}{2}} \bigr\Vert _{0}+ \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert _{0} \bigr) \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert _{1} \leq Ch^{4} \bigl\Vert u^{n-\frac{1}{2}} \bigr\Vert ^{2}_{2}+C \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert ^{2}_{1}. \end{aligned}$$

(25)

Substituting (23)–(25) into (22), we get

$$\begin{aligned} &\bigl\vert \bigl(\vec{f}\bigl(U^{n-\frac{1}{2}}_{h}\bigr)-\vec{f} \bigl(u(t_{n-\frac{1}{2}})\bigr), \nabla \eta ^{n-\frac{1}{2}}\bigr) \bigr\vert \\ &\quad \leq C(\Delta t)^{3} \int ^{t_{n}}_{t_{n-1}} \Vert u_{tt} \Vert ^{2}_{0}\,d\tau +Ch^{4} \bigl\Vert u^{n-\frac{1}{2}} \bigr\Vert ^{2}_{3}+C \bigl\Vert \eta ^{n- \frac{1}{2}} \bigr\Vert ^{2}_{1}. \end{aligned}$$

(26)

Hence, from (18)–(21) and (26), we have

$$\begin{aligned} &\frac{1}{2\Delta t}\bigl[\bigl( \bigl\Vert \eta ^{n} \bigr\Vert ^{2}_{0}- \bigl\Vert \eta ^{n-1} \bigr\Vert ^{2}_{0}\bigr)+ \alpha \bigl( \bigl\vert \eta ^{n} \bigr\vert ^{2}_{1}- \bigl\vert \eta ^{n-1} \bigr\vert ^{2}_{1}\bigr)\bigr] \\ &\quad \leq Ch^{4}+C(\Delta t)^{3} \int ^{t_{n}}_{t_{n-1}}\bigl( \Vert u_{tt} \Vert ^{2}_{1}+ \Vert u_{ttt} \Vert ^{2}_{1}\bigr)\,d\tau +C \bigl\Vert \eta ^{n-\frac{1}{2}} \bigr\Vert _{1}^{2}, \end{aligned}$$

multiplying by $2\Delta t$, then summing up the above inequality and noting that $\eta ^{0}=0$, we can obtain

$$\begin{aligned} \bigl\Vert \eta ^{n} \bigr\Vert ^{2}_{1} \leq Ch^{4}+C(\Delta t)^{4}+C\Delta t\sum ^{n}_{i=1} \bigl\Vert \eta ^{i} \bigr\Vert ^{2}_{1}. \end{aligned}$$

(27)

Choosing Δt small enough so that $1-C\Delta t>0$ and applying discrete Gronwall’s lemma, there holds

$$\begin{aligned} \bigl\Vert \eta ^{n} \bigr\Vert ^{2}_{1}\leq C \bigl(h^{4}+(\Delta t)^{4}\bigr), \end{aligned}$$

the proof is completed. □

To obtain the global superconvergence estimate, we combine the adjacent four small elements $K_{1}$, $K_{2}$, $K_{3}$, $K_{4}$ into a big element K̃, i.e., $\tilde{K}=\bigcup_{i=1}^{4}K_{i}$ (see Fig. 1), the corresponding subdivision is defined by $T_{2h}$.

Figure 1

The big element K̃

As in [19], we define the following interpolation postprocessing operator $I_{2h}$:

$$\begin{aligned} I_{2h}u| _{\tilde{K}}\in Q_{2}(\tilde{K}), \quad \forall u\in C( \tilde{K}), \qquad I_{2h}u(Z_{i})=u(Z_{i}), \quad i=1,2,\ldots ,9, \end{aligned}$$

(28)

where $Q_{2}(\tilde{K})$ and $C(\tilde{K})$ denote the spaces of biquadratic piecewise polynomial and continuous function on K̃, respectively, $Z_{i}$ are all vertices of four small elements (see Fig. 1).

Meanwhile, $I_{2h}$ has the following properties (see [19]):

$$\begin{aligned} I_{2h}I_{h}u=I_{2h}u, \qquad \Vert u-I_{2h}u \Vert _{1}\leq Ch^{2} \Vert u \Vert _{3}, \qquad \Vert I_{2h}v_{h} \Vert _{1}\leq C \Vert v_{h} \Vert _{1}, \quad \forall v_{h}\in V_{h}. \end{aligned}$$

(29)

Theorem 3.2

Under the assumption of Theorem 3.1, there holds the global superconvergence result as follows:

$$\begin{aligned} \bigl\Vert u^{n}-I_{2h}U^{n}_{h} \bigr\Vert _{1}\leq C\bigl(h^{2}+(\Delta t)^{2} \bigr). \end{aligned}$$

Proof

Let $u^{n}-I_{2h}U^{n}_{h}=u^{n}-I_{2h}I_{h}u^{n}+I_{2h}I_{h}u^{n}-I_{2h}U^{n}_{h}$. By (16), (29), and the triangular inequality, we can get

$$\begin{aligned} \bigl\Vert u^{n}-I_{2h}U^{n}_{h} \bigr\Vert _{1}&\leq \bigl\Vert u^{n}-I_{2h}I_{h}u^{n} \bigr\Vert _{1}+ \bigl\Vert I_{2h}I_{h}u^{n}-I_{2h}U^{n}_{h} \bigr\Vert _{1} \\ &\leq \bigl\Vert u^{n}-I_{2h}u^{n} \bigr\Vert _{1}+ \bigl\Vert I_{2h}\bigl(I_{h}u^{n}-U^{n}_{h} \bigr) \bigr\Vert _{1} \\ &\leq Ch^{2} \bigl\Vert u^{n} \bigr\Vert _{3}+C \bigl\Vert I_{h}u^{n}-U^{n}_{h} \bigr\Vert _{1} \\ &\leq C\bigl(h^{2}+(\Delta t)^{2}\bigr), \end{aligned}$$

(30)

the proof is completed. □

Remark 3.1

In this paper, the boundedness of $\|U^{n}_{h}\|_{1}$ is crucial to the unconditional superclose and superconvergence results, the technique we used is more simple and direct than those in [22] and [23] ([22] applied an error splitting technique and [23] employed a complicated mathematical inductive hypothesis method to derive the boundedness of numerical solution).

4 Numerical examples

In this section, we give three examples to verify the validity of theoretical analysis, here we divide the domain Ω into $m\times n$ rectangular meshes.

Example 1

We consider the following homogeneous BBMB equation:

$$\begin{aligned} u_{t}-\Delta u_{t}-\Delta u-\nabla \cdot \vec{f}(u)=0, \quad (x,y,t) \in [0,1]\times [0,1]\times (0,T]. \end{aligned}$$

Initially, the energy $E^{n}=\|U^{n}_{h}\|^{2}_{1}$ under the initial condition $u(x,y,0)=\sin\pi x\sin\pi y$ is plotted in Fig. 2, here $t\in [0, 0.1]$, $h=\frac{1}{20}$, $\Delta t= 1.0\text{e}{-}05$.

Figure 2

$E^{n}$ under the initial condition $u(x,y,0)=\sin\pi x\sin\pi y$

Then a larger initial condition ($u(x,y,0)=100\sin\pi x\sin\pi y$) is considered, and $E^{n}$ is displayed in Fig. 3, here t, h, and Δt are the same as above.

Figure 3

$E^{n}$ under the initial condition $u(x,y,0)=100\sin\pi x\sin\pi y$

From Figs. 2–3 we can see that the energy is stable, which is consistent with the conclusion of Theorem 2.1.

Example 2

We consider the following inhomogeneous BBMB equation:

$$\begin{aligned} \textstyle\begin{cases} u_{t}-\Delta u_{t}-\Delta u-\nabla \cdot \vec{f}(u)=g(x,y,t), & (x,y,t) \in [0,1]\times [0,1]\times (0,T], \\ u(x,y,0)=2xy(x-1)(y-1), & (x,y)\in [0,1]\times [0,1], \end{cases}\displaystyle \end{aligned}$$

here $g(x,y,t)$ could be computed by the exact solution $u(x,y,t)=(1+e^{-t})xy(x-1)(y-1)$.

First of all, we take $\|I_{h}u^{n}-U^{n}_{h}\|_{1}$ as an example to validate the unconditional stability. Here fix $h=\frac{1}{100}$ and choose $\Delta t= \frac{h}{10},\frac{h}{20},\frac{h}{40},\frac{h}{80}$, respectively, we provide the results of $\|I_{h}u^{n}-U^{n}_{h}\|_{1}$ at $t=0.1$, 0.5, and 1.0 in Table 1.

Table 1 Error results of $\|I_{h}u^{n}-U^{n}_{h}\|_{1}$ with $h=\frac{1}{100}$

From Table 1, we can observe that $\|I_{h}u^{n}-U^{n}_{h}\|_{1}$ is stable at a certain time and free from the ratio between Δt and h.

Moreover, the convergence, superclose, and superconvergence results at $t=0.1$, 0.5, and 1.0 are listed in Tables 2–4, respectively. At the same time, we describe the error reduction results in Figs. 4–6, respectively. To confirm the convergence order, we choose $\Delta t=h$.

Figure 4

Numerical results for u at $t=0.1$

Figure 5

Numerical results for u at $t=0.5$

Figure 6

Numerical results for u at $t=1.0$

Table 2 Numerical results of u at $t=0.1$

Table 3 Numerical results of u at $t=0.5$

Table 4 Numerical results of u at $t=1.0$

From Tables 2–4 and Figs. 4–6, we can see that $\|u^{n}-U^{n}_{h}\|_{1}$ is convergent at rate of $O(h)$, $\|I_{h}u^{n}-U^{n}_{h}\|_{1}$ and $\|u^{n}-I_{2h}U^{n}_{h}\|_{1}$ are convergent at rate of $O(h^{2})$, which coincides with the conclusions of Theorems 3.1–3.2.

Example 3

We consider the inhomogeneous BBMB equation in larger domains.

Firstly, we introduce the following equation:

$$\begin{aligned} \textstyle\begin{cases} u_{t}-\Delta u_{t}-\Delta u-\nabla \cdot \vec{f}(u)=g(x,y,t), & (x,y,t) \in [0,5]\times [0,5]\times (0,T], \\ u(x,y,0)=2xy(x-5)(y-5), & (x,y)\in [0,5]\times [0,5], \end{cases}\displaystyle \end{aligned}$$

where the exact solution $u(x,y,t)=(1+e^{-t})xy(x-5)(y-5)$.

Here we take $t=1, 2$, and 3, respectively, the unconditional stability is validated in Table 5, the convergence, superclose, and superconvergence results are listed in Tables 6–8, respectively.

Table 5 Error results of $\|I_{h}u^{n}-U^{n}_{h}\|_{1}$ with $h=\frac{1}{20}$

Table 6 Numerical results of u at $t=1$

Table 7 Numerical results of u at $t=2$

Table 8 Numerical results of u at $t=3$

Secondly, we consider the following equation:

$$\begin{aligned} \textstyle\begin{cases} u_{t}-\Delta u_{t}-\Delta u-\nabla \cdot \vec{f}(u)=g(x,y,t), & (x,y,t) \in [0,8]\times [0,8]\times (0,T], \\ u(x,y,0)=2xy(x-8)(y-8), & (x,y)\in [0,8]\times [0,8], \end{cases}\displaystyle \end{aligned}$$

here $u(x,y,t)=(1+e^{-t})xy(x-8)(y-8)$.

The convergence, superclose, and superconvergence results at $t=1, 2$, and 3 are provided in Tables 9–11, respectively.

Table 9 Numerical results of u at $t=1$

Table 10 Numerical results of u at $t=2$

Table 11 Numerical results of u at $t=3$

From Tables 5–11, we can see that under the large initial condition, the numerical results are also in good agreement with our theoretical analysis.