Superconvergence analysis of an energy stable scheme for nonlinear reaction-diffusion equation with BDF mixed FEM
Introduction
In this paper, the following nonlinear reaction-diffusion equation with homogeneous boundary condition is considered: Assume is a rectangle with the boundary ∂Ω, , and .
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 () unconditionally. Inspired by [5], [6], in order to ensure the energy stability, an additional term is added to the approximation scheme. Based on the energy stability, the boundedness of the numerical solutions about in -norm and in -norm become available for . The error and are split into the temporal errors and the spatial errors by a time-discrete system with solutions and . The monotonicity is utilized to derive the temporal error for in -norm and in -norm. The boundedness of is gained unconditionally by spatial error in -norm with order , where is the associated interpolation operator. With the help of the boundedness of and the connection between and , superclose properties of with order in -norm and with order in -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 plane with edges parallel to the coordinate axes and be a regular rectangular subdivision. For given , let the four vertices and edges are and , respectively. The associated FE spaces and are defined by and respectively. Let and are the interpolation operators satisfying and , where 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: We get by the following equations: and where and . We set and . Further, we denote , , and
The spatial error analysis
In this section, we will give the error estimates between and in -norm, which can be used to get the boundedness of unconditionally. Denote and Theorem 4 Let and be the solutions of (2.5) and (2.8)–(2.10) respectively, for , under the conditions of Theorem 2, there
Unconditional superconvergence results for fully discrete system
Theorem 5 Let and be the solutions of (2.5) and (2.8)–(2.10) respectively, for , under the conditions in Theorem 3, we have Proof By the help of Theorem 3 and Theorem 4, we have and We proceed with (5.1) for . Setting in the first equation of (4.7) and in the second equation of (4.7), we derive
Numerical results
In this section, we consider the nonlinear reaction-diffusion equation: with , . We divide Ω into rectangles. In our computation, rectangular partitions with and 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 . Example 1 is chosen corresponding to the exact solution .
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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