Superconvergence analysis of a MFEM for BBM equation with a stable scheme

doi:10.1016/j.camwa.2021.04.016

Computers & Mathematics with Applications

Volume 93, 1 July 2021, Pages 168-177

https://doi.org/10.1016/j.camwa.2021.04.016 Get rights and content

Abstract

In this paper, superconvergence properties are presented for the nonlinear Benjamin-Bona-Mahony (BBM) equation with a mixed finite element method (MFEM). We propose an Euler scheme combined with the artificial Douglas-Dupont regularization terms, which guarantee the stability of the numerical scheme. Splitting technique is utilized to get rid of the ratio between the time step τ and the subdivision parameter h. Temporal error estimates and unconditional spatial error estimates are gained through some techniques, such as the function's monotonicity and the Green formula and so on. In turn, the regularities of the solutions about the time-discrete equations and the boundedness of the numerical solution are derived. Based on the above achievements, the unconditional superconvergent results of $u^{n}$ in $H^{1}$ -norm and ${\vec{q}}^{n}$ in $L^{2}$ -norm with order $O (h^{2} + τ)$ are obtained through the trigonometric inequality. The global superconvergent results are deduced by use of the interpolated postprocessing operators. Numerical example shows the validity of the theoretical analysis.

Introduction

In 1972, the Benjamin-Bona-Mahony (BBM) equation was put forward by Benjamin, Boehner and Mahoney when simulating small amplitude long wave in fluid mechanics. This equation is also known as the regularized long-wave equation, which is an important nonlinear partial differential equation. It is widely used in chemistry, physics, biology and many other fields, so it is necessary to study the numerical methods of BBM equation. Numerical solutions of the BBM equation in one space dimension were discussed in [1] and [2]. The Crank-Nicolson-type finite difference method was considered in [1], [2] and both of them studied the existence, stability and uniqueness of the numerical equations. Further, convergent results in $H^{1}$ -norm and $L^{\infty}$ -norm were given in [1] and [2], respectively. A linear triangular finite element method (FEM) was applied to the nonlinear BBM equation on anisotropic meshes in [3]. Through the relationship between the interpolation and the Ritz projection, the superconvergence results were obtained. In [4] and [5], the two-grid FEM was utilized, and both of them gave the superconvergence analysis for the nonlinear BBM equation. The average vector field method and the implicit midpoint method were used to derive the temporal discretization and spatial discretization, respectively in [6]. And the local mass and local energy were conserved by the algorithm. A mixed finite element method (MFEM) was proposed for solving the nonlinear BBM equation in [7] and the superclose properties and the global superconvergence results were arrived at.

In fact, as an important numerical method, MFEM has been widely employed in partial differential equations for its simplicity and high efficiency [8], [9]. However, in the traditional MFEM, choosing the two spaces which require to satisfy a LBB stability condition is not a simple thing. In simulating second order elliptic problem, the authors showed a new MFEM that the two approximation spaces are much easier to be selected to preserve the LBB stability condition in [10] and [11]. Subsequently, such method was also applied to many other nonlinear evolution equations [7], [12], [13], [14].

However, studying a nonlinear evolution equation by virtue of the FEM always causes the boundedness of the numerical solution in some norms. As a result, the restriction of the ratio between the time step τ and the subdivision parameter h is inevitable in the classical way. For the sake of getting rid of such restriction, a fascinating insight which is called as splitting technique was displayed in [15], [16], [17], [18], [19] and unconditional optimal error estimates with conforming FEM were deduced in all of the these literature. Later, [20], [21], [22], [23], [24] perform unconditional superconvergence behavior of some other nonlinear evolution equations by use of such technique.

On the other hand, as we can see that, the conservative schemes perform better than the nonconservative ones for FEM. Furthermore, the schemes preserving in mass or energy conservation have been extensive researched for nonlinear evolution equations [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. Specially, an artificial Douglas-Dupont regularization term was added in the numerical scheme for the Cahn-Hilliard equation to derive the energy stability in [36] and optimal error estimate was showed. Also, such technique was applied to nonlinear reaction-diffusion equation in [14] for energy stability and superconvergent results were obtained.

In this paper, we are concerned the following nonlinear BBM equation: ${\begin{matrix} u_{t} - Δ u_{t} = \nabla \cdot \vec{f} (u), & (X, t) \in Ω \times (0, T], \\ u (X, t) = 0, & (X, t) \in \partial Ω \times (0, T], \\ u (X, 0) = u_{0} (X), & X \in \bar{Ω} . \end{matrix}$ Suppose $Ω \subset R^{2}$ is a rectangle with the boundary ∂Ω, $0 < T < \infty$ , $X = (x, y)$ and $\vec{f} (u) = (- \frac{1}{2} u^{2} - u, - \frac{1}{2} u^{2} - u)$ . Motivated by [14], [36], we establish unconditional superconvergence estimates of a nonlinear iteration scheme for (1.1) with MFEM, which improves the results of the existing literature. Firstly, an Euler scheme combined with artificial Douglas-Dupont regularization terms $τ ({(U_{h}^{n})}^{3}, v_{h})$ and $τ (U_{h}^{n}, v_{h})$ is developed for the nonlinear BBM equation, where $U_{h}^{n}$ is the solution of the numerical equation and $v_{h}$ is the trial function in the FE space. We take advantage of the regularization terms and Green formula to guarantee the stability of the numerical scheme. Secondly, by the means of splitting technique, we get rid of the ratio between τ and h. We add the terms $τ {(U^{n})}^{3}$ and $τ U^{n}$ into the time-discrete system to consist with the numerical equations. To avoid the difficulties brought by $τ {(U^{n})}^{3}$ , the function's monotonicity is utilized to derive the temporal error $e^{n}$ in $H^{1}$ -norm. In what follows, we analyze the temporal estimate of $e^{n}$ in $H^{2}$ -norm. Based on the temporal errors, the spatial errors of $I_{h} U^{n} - U_{h}^{n}$ in $H^{1}$ -norm and $Π_{h} {\vec{Q}}^{n} - {\vec{Q}}_{h}^{n}$ in $L^{2}$ -norm are displayed with order $O (h (h + τ))$ , where $I_{h}$ and $Π_{h}$ are the corresponding interpolation operators. Accompanied by the above results, unconditional boundedness of ${‖ U_{h}^{n} ‖}_{0, \infty}$ is obtained. Thirdly, we get unconditional superclose properties of $U_{h}^{n} - I_{h} u^{n}$ in $H^{1}$ -norm and ${\vec{Q}}_{h}^{n} - Π_{h} {\vec{q}}^{n}$ in $L^{2}$ -norm through the trigonometric inequality. At last, we discuss the global superconvergence by using the postprocessing operators in [37]. A numerical experiment has been conducted to validate the theoretical results.

Section snippets

An energy stable scheme with MFEM

We start by section 2 with the MFE scheme, which is energy stable. Assume that the edges of Ω are parallel to the coordinate axes and $Γ_{h}$ is a regular rectangular subdivision. Meanwhile, we define the FE spaces $V_{h}$ and ${\vec{W}}_{h}$ by $V_{h} = {v; v |_{K} \in Q_{11}, \forall K \in Γ_{h}, v |_{\partial Ω} = 0}$ and ${\vec{W}}_{h} = {\vec{w} = (w^{1}, w^{2}); \vec{w} |_{K} \in Q_{01} \times Q_{10}, \forall K \in Γ_{h}} .$ Let $I_{h}$ and $Π_{h}$ are the associated interpolation operators, respectively. The following results, which are necessary to get the spatial error estimates, can be found in [12], [37]. When $u \in H^{3} (Ω)$ and $\vec{q} \in {(H^{2} (Ω))}^{2}$ ,

The temporal error analysis

As a preparatory work, section 3 is devoted to construct a time-discrete system, which frees the superconvergent results from the restriction of τ in the next section. The time-discrete system is given as follows: $D_{τ} U^{n} - D_{τ} Δ U^{n} - \nabla \cdot \vec{f} (U^{n}) + τ {(U^{n})}^{3} + τ U^{n} = 0,$ where $U^{0} = u_{0} (X)$ and $U^{n} |_{\partial Ω} = 0$ . Further, we employ ${\vec{Q}}^{n} = D_{τ} \nabla U^{n} + \vec{f} (U^{n})$ and denote $e_{u}^{n} \dot{=} u^{n} - U^{n}, {\vec{e}}_{q}^{n} \dot{=} {\vec{q}}^{n} - {\vec{Q}}^{n}, n = 1, 2, \dots, N$ . Before starting the temporal error estimates, we have a brief pause for a lemma, which can be found in [38], [39].

Lemma 1

(the Gronwall inequality)

The spatial error analysis and superconvergence results

According to those preparations above, we turn to the investigation of unconditional spatial error estimates in this section. Finally, we obtain superconvergence results through trigonometric inequality with the spatial error results. Denote $U^{i} - U_{h}^{i} = U^{i} - I_{h} U^{i} + I_{h} U^{i} - U_{h}^{i} \dot{=} η^{i} + ξ^{i}, i = 0, 1, 2, \dots N,$ and ${\vec{Q}}^{i} - {\vec{Q}}_{h}^{i} = {\vec{Q}}^{i} - Π_{h} {\vec{Q}}^{i} + Π_{h} {\vec{Q}}^{i} - {\vec{Q}}_{h}^{i} \dot{=} {\vec{r}}^{i} + {\vec{θ}}^{i}, i = 0, 1, 2, \dots N .$

Theorem 3

Let ${u, \vec{q}}$ and ${U_{h}^{n}, {\vec{Q}}_{h}^{n}}$ be the solutions of (2.5) and (2.6a), (2.6b) respectively, for $n = 1, 2, \dots, N$ , there exist small enough $h > 0$ and $τ > 0$ , such that ${‖ \nabla ξ^{n} ‖}_{0} = O (h (h + τ)$

Numerical simulations

In this section, we consider the nonlinear BBM equation: ${\begin{matrix} u_{t} - Δ u_{t} = \nabla \cdot f (u) + g (X, t), & (X, t) \in Ω \times (0, T], \\ u (X, t) = 0, & (X, t) \in \partial Ω \times (0, T], \\ u (X, 0) = u_{0} (X), & X \in \bar{Ω}, \end{matrix}$ with $\bar{Ω} = [0, 1] \times [0, 1]$ . A uniform rectangular partition with $m + 1$ nodes in each direction is used in our computation. We solve the system by MFEM with $\vec{q} = \nabla u_{t} + \vec{f} (u)$ . We select the FE pair of $Q_{11} + Q_{01} \times Q_{10}$ and $g (X, t)$ is chosen corresponding to the exact solution $u = e^{t} \sin π x \sin π y$ . Furthermore, $\vec{f} (u) = (- \frac{1}{2} u^{2} - u, - \frac{1}{2} u^{2} - u)$ .

We test the results with $τ = h^{2}$ , and the numerical

Acknowledgements

This work was supported by Training Plan of Young Backbone Teachers in Colleges of Henan Province (No. 2020GGJS230), NSF of China (No. 11801527), and China Postdoctoral Science Foundation (No. 2018M632791).

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Superconvergence analysis of a MFEM for BBM equation with a stable scheme

Abstract

Introduction

Section snippets

An energy stable scheme with MFEM

The temporal error analysis

The spatial error analysis and superconvergence results

Numerical simulations

Acknowledgements

Appl. Math. Comput.

Appl. Math. Lett.

Appl. Numer. Math.

Appl. Math. Comput.

Appl. Numer. Math.

Appl. Math. Comput.

Appl. Math. Comput.

J. Comput. Phys.

Appl. Math. Comput.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Appl. Math.

Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation

Numer. Methods Partial Differ. Equ.

A new two-grid nonconforming mixed finite element method for nonlinear Benjamin-Bona-Mahoney equation

Appl. Math. Comput.

A new low order mixed FEM for nonlinear Benjamin-Bona-Mahony equation

Math. Appl.

Mixed element algorithm based on a second-order time approximation scheme for a two-dimensional nonlinear time fractional coupled sub-diffusion model

Eng. Comput.

A coupling method based on new MFE and FE for fourth-order parabolic equation

J. Appl. Math. Comput.

New mixed finite schemes for second order elliptic problem

Math. Numer. Sin.

High accuracy analysis of new schemes for second order elliptic problem for recurrent event data

Acta Math. Appl. Sin.