Elsevier

Applied Numerical Mathematics

Volume 153, July 2020, Pages 457-472
Applied Numerical Mathematics

Superconvergence analysis of an energy stable scheme for nonlinear reaction-diffusion equation with BDF mixed FEM

https://doi.org/10.1016/j.apnum.2020.03.007Get rights and content

Abstract

A step-2 backward differential formula (BDF) temporal discretization scheme is constructed for nonlinear reaction-diffusion equation and superconvergence results are studied by mixed finite element method (FEM) with the elements Q11 and Q01×Q10 unconditionally. In particular, we apply an artificial regularization term to guarantee the energy stability of the step-2 BDF scheme. Splitting technique is utilized to get rid of the ratio between the time step size τ and the subdivision parameter h. Temporal error estimates in H2-norm are derived by use of the function's monotonicity, which leads to the regularities of the solutions for the time-discrete equations. Spatial error estimates in L2-norm are deduced to bound the numerical solution in L-norm. Unconditional superconvergence estimates of un in H1-norm and qn=un in (L2)2-norm with order O(h2+τ2) are obtained. The global superconvergent properties are deduced through above results. Two numerical examples testify the theoretical analysis.

Introduction

In this paper, the following nonlinear reaction-diffusion equation with homogeneous boundary condition is considered:{utΔu+f(u)=0,(X,t)Ω×(0,T],u=0,(X,t)Ω×[0,T],u(X,0)=u0(X),XΩ. Assume ΩR2 is a rectangle with the boundary ∂Ω, 0<T<, X=(x,y)Ω and f(u)=u3u.

Nonlinear reaction diffusion equation has attracted the attention of many scholars and experts [29], [42], [40], [27] for a long time. Particularly, numerical methods such as linearization method in [28], moving grid FEM in [26], expanded mixed FEM in [25], have already been applied to the convergence analysis of nonlinear reaction diffusion equation.

Mixed FEM is an important numerical method and has been widely used in evolution equations. However, for the FE approximation spaces must meet the LBB condition in the conventional mixed FEM, the appropriate FE pairs are not easy to find. In order to resolve this difficulty, [3] and [30] discussed a new mixed FEM for second order elliptic problem and the two approximation spaces only require satisfying a simple inclusive relation. Subsequently, such method was also utilized to many other PDEs, for instance, the Sobolev equations [35], the parabolic problems [34], the BBM equations [37], and so on.

On the other hand, to study the time-dependent optimal error estimates for a nonlinear physical system, the boundedness of numerical solution in some norms is often needed. Due to the inverse inequality, employed to deal with such issue, the ratio between the time step size and the subdivision parameter is inevitable by the normal method. In [16], [39], [17], [36], [31], [32], [33], [38], the splitting technique was used for getting rid of the restriction of such ratio. Especially for BDF schemes [12], [15], [24], [2], [10] and [1] applied splitting technique to time-dependent nonlinear thermistor equation and nonlinear Schrödinger equation with a step-2 BDF scheme and a step-3 BDF scheme, respectively. Convergence results have been acquired unconditionally in [10], [1].

Moreover, the authors in [5], [6] also constructed a step-2 BDF scheme for Cahn-Hilliard equation and SPFC equation, respectively. In order to derive the energy stability for the numerical scheme, a second order artificial Douglas-Dupont regularization term was utilized and optimal rate convergence results were obtained. From the numerical point of view, the nonconservative methods may easily show nonlinear blow-up and the conservative schemes are much more popular. In addition, the schemes preserving in mass or energy conservation have been studied extensively for some corresponding equations [41], [11], [19], [13], [18], [20], [22], [21]. In fact, there have been quite a few recent works [14], [8], [4], [7], [9] for the second order accurate, energy stable numerical schemes and the unique solvability, energy stability and optimal rate convergence analysis have been presented in these works. However unconditional superconvergence analysis has not been researched further.

In our paper, we develop step-2 BDF mixed FE scheme for (1.1) and discuss superconvergence estimates with the conforming FE pair (Q11+Q01×Q01) unconditionally. Inspired by [5], [6], in order to ensure the energy stability, an additional term Aτ(unun1),(A14) is added to the approximation scheme. Based on the energy stability, the boundedness of the numerical solutions about Uhn in H1-norm and Qhn in (L2)2-norm become available for n2. The error unUhn and qnQhn are split into the temporal errors and the spatial errors by a time-discrete system with solutions Un and Qn. The monotonicity is utilized to derive the temporal error for eun in H2-norm and eqn in (H1)2-norm. The boundedness of Uhn0, is gained unconditionally by spatial error IhUnUhn in L2-norm with order O(h(h+τ)), where Ih is the associated interpolation operator. With the help of the boundedness of Uhn0, and the connection between Dτσn and ¯tσn, superclose properties of UhnIhun with order O(h2+τ2) in H1-norm and QhnΠhqn with order O(h2+τ2) in (L2)2-norm are deduced unconditionally. Finally, global superconvergence results are derived by use of the postprocessing operators in [35]. Two numerical examples are given to confirm the validity of the theoretical analysis.

Section snippets

An energy stable and fully-discrete mixed FE approximation scheme

Let Ω be a rectangle in (x,y) plane with edges parallel to the coordinate axes and Γh be a regular rectangular subdivision. For given KΓh, let the four vertices and edges are ai,i=14 and li=aiai+1,i=14(mod4), respectively. The associated FE spaces Vh and Wh are defined byVh={v;v|KQ11,KΓh,v|Ω=0} andWh={w=(w1,w2);w|KQ01×Q10,KΓh}, respectively. Let Ih and Πh are the interpolation operators satisfying Ihu(ai)=u(ai) and li(Πhqq)sds=0,i=14, where s is the tangent vector. It can

The temporal error analysis

In this section, our purpose is to split the error by constructing a time-discrete system, which is the basis on getting rid of the restriction of h and τ. Now the time-discrete system is given as follows:DτUnΔUn+f(Un)+Aτ(UnUn1)=0,n2. We get U1 by the following equations:U1U0τΔU1+ΔU02+f(U1,0+U02)=0, andU1,0U0τΔU1,0+ΔU02+f(U0)=0, where U0=u0(X) and U1,0|Ω=0,U1|Ω=0. We set Q1,0=U1,0 and Qn=Un,n=1,2,,N. Further, we denote eu1,0=˙u1,0U1,0, eq1,0=˙q1,0Q1,0, eun=˙unUn and eqn=˙q

The spatial error analysis

In this section, we will give the error estimates between IhUn and Uhn in L2-norm, which can be used to get the boundedness of Uhn0, unconditionally. DenoteU1,0Uh1,0=U1,0IhU1,0+IhU1,0Uh1,0=˙η1,0+ξ1,0,Q1,0Qh1,0=Q1,0ΠhQ1,0+ΠhQ1,0Qh1,0=˙r1,0+θ1,0,UiUhi=UiIhUi+IhUiUhi=˙ηi+ξi,i=0,1,2,N, andQiQhi=QiΠhQi+ΠhQiQhi=˙ri+θi,i=0,1,2,N.

Theorem 4

Let {u,q} and {Uhn,Qhn} be the solutions of (2.5) and (2.8)(2.10) respectively, for n=1,2,,N, under the conditions of Theorem 2, there

Unconditional superconvergence results for fully discrete system

Theorem 5

Let {u,q} and {Uhn,Qhn} be the solutions of (2.5) and (2.8)(2.10) respectively, for n=1,2,,N, under the conditions in Theorem 3, we have(IhunUhn)0+ΠhqnQhn0=O(h2+τ2).

Proof

By the help of Theorem 3 and Theorem 4, we have(Ihu1Uh1)0(Iheu1eu1)0+eu10+(IhU1Uh1)0C(h2+τ2), andΠhq1Qh10Πheq1eq10+eq10+ΠhQ1Qh10C(h2+τ2). We proceed with (5.1) for n2. Setting vh=Dτξn in the first equation of (4.7) and wh=Dτξn in the second equation of (4.7), we derive{(Dτξn,Dτξn)

Numerical results

In this section, we consider the nonlinear reaction-diffusion equation:{utΔu+f(u)=g(X,t),(X,t)Ω×(0,T],u=0,(X,t)Ω×(0,T],u(X,0)=u0(X),XΩ, with Ω=[0,1]×[0,1], f(u)=u3u. We divide Ω into m×n rectangles. In our computation, rectangular partitions with m+1 and n+1 nodes in the two different direction are used, respectively. We solve the system by the step-2 BDF mixed FEM with the FE pair of Q11+Q01×Q10.

Example 1

g(X,t) is chosen corresponding to the exact solution u=etxy(1x)(1y).

To confirm the error

Acknowledgements

This work was supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 20A110030), the Doctoral Starting Foundation of Pingdingshan University (No. PXY-BSQD-2019001) and the University Cultivation Foundation of Pingdingshan (No. PXY-PYJJ-2019006).

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