The Theory and Computation of the Semi-Linear Reaction–Diffusion Equation with Dirichlet Boundaries

Abstract

In this article, we study the semi-linear two-dimensional reaction–diffusion equation with Dirichlet boundaries. A reliable numerical scheme is designed, coupling the nonstandard finite difference method in the time together with the Galerkin in combination with the compactness method in the space variables. The aforementioned equation is analyzed to show that the weak or variational solution exists uniquely in specified space. The a priori estimate obtained from the existence of the weak or variational solution is used to show that the designed scheme is stable and converges optimally in specified norms. Furthermore, we show that the scheme preserves the qualitative properties of the exact solution. Numerical experiments are presented with a carefully chosen example to validate our proposed theory.

1. Introduction

Many physical and mathematical developments that have taken place recently (due to various chemical as well as diffusion-like systems) in the sciences, ecology, population dynamics, biochemistry and engineering, to mention a few fields, have attracted extensive attention. These physical developments correspond to several phenomena in which chemical substances are transformed into each other. For more on these, see [1,2]. All these activities are modeled to mathematical equations describing semi-linear parabolic partial differential equation, where the reaction–diffusion equation occurs. In this paper, we are concerned with the study of the following two-dimensional semi-linear reaction–diffusion equation of the following type:

$$\begin{array}{c}\hfill \frac{\mathsf{\partial }u}{\mathsf{\partial }t}-\Delta u+{\left(\right|u|}^2-1)u=0,(x,t)\in \Omega \times (0,T)\end{array}$$

(1)

$$\begin{array}{c}\hfill u(x,t)=0\mathrm{on}\mathsf{\partial }\Omega \times (0,T)\end{array}$$

(2)

$$\begin{array}{c}\hfill u(x,0)=u_0\left(x\right)\mathrm{for}x\in \Omega t=0,\end{array}$$

(3)

where $\Omega \subset {\mathbb{R}}^2$ is an open-bounded set contained in ${\mathbb{R}}^2$ with D as a matrix with positive constant and $u:\Omega ⟶\mathbb{R}$ as an unknown scalar function.

In addition to the description of the formation of the semi-linear reaction–diffusion equation stated above, these equations originate naturally in systems consisting of many interesting components in a variety of biological, chemical, physical and population dynamics fields. In certain situations, as in [2], the patterns are seen as phase-field transitional systems. This is a situation in which the phase function u describes the transition between the solid and liquid phases in the solidification process of a material occupying a region $\Omega$. The above reaction–diffusion equation could be seen to be exhibiting a type of Allen–Cahn equation in [3,4,5], which is used to describe the motion of anti-phase boundaries in crystalline solids. This has been widely applied to many complex moving interface problems, for example the mixture of two incompressible fluids and the nucleation of solids and vesicle membranes; see [6] for more. Specifically, the Equations (1)–(3) arise in the study of the superconductivity of liquids and have been studied in J. Smoller [7]. A great deal of work has been undertaken on the nonlinear reaction–diffusion equation, and for more on this see [8,9,10,11,12].

The above class of partial differential equations have been analyzed using several methods, such as the fixed node finite difference methods in [13] and the spectral method in [14]. This was followed by some other methods, which have good numerical stability and can be used for multi-dimensional purposes, as presented in [15]. These methods include the method by Sharifi and Rashidian [16], who applied an explicit finite difference associated with extended cubic B-spline collocation method. There is also a method by Wang et al. [17] called the compact boundary value method (CBVM), which is a combination of a compact fourth-order differential method (CFODM) and the P-order boundary value method (POBVM). This method is locally stable and when applied to the equation will obtain a unique solution. Another method is the one by Wu et al. [18], called the variational iteration method (VIM), used for structuring equations, and the Lagrange multipliers and numerical integral formulas are used with this method. Last but not the least is the method by Biazar and Mehriatifan in [19], which solved the equation using the compact finite difference method.

In contrast to the above good methods stated for the analysis of the problem, only very few or none have used the coupling technique as a tool for the analysis of this problem, to the best of this author’s knowledge. To this end, we exploited the gap and designed a very reliable and efficient scheme, consisting of the nonstandard finite difference method in time and the Galerkin combined with the compactness method in the space variables (NSFD-GM) in a two-dimensional setting. This method can be used to show that the solution obtained from the semi-linear reaction–diffusion equation exists uniquely. We also proceed to show, using the a priori estimates obtained from the existence of the variational or weak solution, that the designed scheme is stable. The stability of the scheme is further tested for its optimal convergence with the $L^2$- as well as the $H^1$-norms to be defined later. Furthermore, we show that the designed scheme (NSFD-GM) preserves or replicates the decaying properties of the exact solution. Numerical experiments are then presented with a carefully chosen example to justify that the numerical results obtained from the above experiments indeed validate the theory presented in the work.

The above method has been used extensively in a one-dimensional situation such as in [20]. It should, however, be noted that the aforementioned one-dimensional case was an extension of the pioneering work in [21], from where the method was initiated in a linear situation over a non-smooth geometry. Our goal in this article was to see whether or not the method could be applicable in a two- or higher-dimensional setting. The relevance of the method could be deduced from the fact that the designed schemes that have in the past emanated from this method preserve or replicate all the qualitative properties of the exact solution of the problems that they have solved. In addition, where the method has been used, it has also performed better than the traditional Euler method. One could also attempt to remark that since real life problems such as this appear mostly in higher dimensional settings, if the method is found to perform exceptionally well, then that will be a remarkable achievement for the method in that framework. The founder of the NSFD method was Mickens [22] and major contributions to the foundation and application to the method can be found in [23,24,25,26]. As for the overview of the method, see [27]. The comparison of the finite and nonstandard finite difference methods can be seen in [22].

The organization of this article begins from Section 2, where we briefly outline the notations and tools to be used subsequently in the article. This is followed by Section 3, which addresses the existence and uniqueness of the solution of the problem. We proceed to design the numerical scheme in which convergence analysis is established in Section 4. Section 5 is then devoted to carrying out the numerical experiments with a carefully chosen example to justify the theory. Conclusions and future remarks are finally stated in Section 6.

2. Notations and Preliminaries

In this section, for the sake of completeness, we gather some important notations with tools such as definitions, concepts and inequalities that will play some important roles in the analysis of the problem. Some of these preliminary tools may duplicate some tools from these papers [20,28,29,30], the reason being that these papers deal with related subjects that need the same tools. Among these tools are function spaces such as the space $\mathcal{D}(\Omega )$. This space, in summary, is defined as a linear space of infinitely differentiable functions with compact support in the domain $\Omega$, where $\Omega$ in our case will be a two-dimensional open set contained in ${\mathbb{R}}^2$. The above mentioned space $\mathcal{D}(\Omega )$ is followed by the space of distributions ${\mathcal{D}}^{\prime }(\Omega )$, which is its pair. The duality pairing of these two spaces is denoted by $〈·,·〉$. In between these two spaces are the $L^p(\Omega )$ spaces which are also essential, and this, in summary, is defined by

$$\begin{array}{c}\hfill L^p(\Omega ):=\left\{v:{\left({\int }_{\Omega }{\left|v\left(x\right)\right|}^{p}dx\right)}^{1/p}<\infty \right\}.\end{array}$$

for $1<p<+\infty$. The $L^p(\Omega )$ space is known to be a Banach space with the norm defined by

$$\begin{array}{c}\hfill {\parallel v\parallel }_{L^p(\Omega )}={\left({\int }_{\Omega }{\left|v\left(x\right)\right|}^{p}dx\right)}^{1/p}.\end{array}$$

(4)

For more on these spaces, see [31,32,33]. Another very important function space is the Sobolev space, denoted and defined by

$$\begin{array}{c}\hfill W^{m,p}(\Omega ):=\left\{v\in L^p(\Omega ):D^{\alpha }v\in L^p(\Omega ),\mathrm{for} \mathrm{all} \mathrm{multi} \mathrm{index}\left|\alpha \right|\le m\right\}.\end{array}$$

(5)

for $m\in \mathbb{N}$ and $p\in \mathbb{R}$ with $1<p\le \infty$. The above space (5) with the norm defined by

$$\begin{array}{c}\hfill {\parallel v\parallel }_{m,p,\Omega }={\left(\sum _{\left|\alpha \right|\le m}{\parallel D^{\alpha }v\parallel }_{L^p(\Omega )}\right)}^{1/p},p<\infty \end{array}$$

(6)

or

$$\begin{array}{c}\hfill {\parallel v\parallel }_{m,\infty ,\Omega }=\underset{\left|\alpha \right|\le m}{sup}\left(ess\underset{x\in \Omega }{sup}\left|D^{\alpha }v\left(x\right)\right|\right),p=\infty .\end{array}$$

(7)

which is also a Banach space. When $p=2$ in the above Sobolev space, the notation $W^{m,2}(\Omega )$ is applied and often denoted by $H^m(\Omega )$ and called a Hilbert space. Most often, we use the superscript $p=2$ when referring to its norm and semi-norm, especially where there is no ambiguity. See Lions and Magenes [33] for more details. Continuing in the gathering of tools, we would like to introduce another, more general Sobolev space denoted by $H^m[(0,T);X]$, where X is a Hilbert space. The above space, as viewed by Lions and Magenes [33], is defined in summary as the space of square integrable functions taking values from $[0,T]$ to X. The norm of this space is defined by

$$\begin{array}{c}\hfill {\parallel v\parallel }_{H^m[(0,T);X]}:={\left(\sum _{\left|\alpha \right|\le m}{\int }_0^T{\parallel D^{\alpha }v\left(x\right)\parallel }_X^{2}dt\right)}^{1/2}.\end{array}$$

(8)

In view of (8), X could be seen as either $L^p$ or $W^{m,p}$ space and in a particular situation of ours, $H_0^1,L^2,L^4$ and $H_0^m$. The following inequalities will be used in this paper: Hölder, Gronwall’s, Young’s, Poincaré and Gagliador-Nirenburg inequality. For more inequalities not listed here, we will refer to standard reference materials such as in [31,32,33,34,35] when required.

We conclude the assembly of the tools by introducing the following numerical space, denoted by ${\mathcal{V}}_h$, needed for the analysis of the discrete problem. This space is finite-dimensional and is defined by

$$\begin{array}{c}\hfill {\mathcal{V}}_h:=\left\{v_h\in C^0\left(\overline{\Omega }\right):v_h{{|}_{\mathsf{\partial }\Omega }=0,v_h|}_{\mathcal{J}}\in P_1,\forall \mathcal{J}\in {\mathcal{J}}_h\right\}\end{array}$$

(9)

where $P_1$ is the space of a polynomial of a degree less than or equal to 1. ${\mathcal{J}}_h$, as stated above, will be a regular mesh of the domain $\Omega$, consisting of compatible triangles $\mathcal{J}$ of sizes $h_{\mathcal{J}}<h$. For more details on this see [34]. In view of (9), we observe that, for each ${\mathcal{J}}_h$, we associate a finite element space $\mathcal{V}$ of a continuous linear piece-wise test function with a value of 1 or zero at every other node of $\mathcal{V}$. That is, if ${\left\{P_j\right\}}_{j=1}^n$ are the interior nodes of ${\mathcal{J}}_h$, then any function in ${\mathcal{V}}_h$ is uniquely determined by its values at the point $P_j$ and it should also be noted that ${\mathcal{V}}_h\subset H_0^1(\Omega )$.

3. The Semi-Linear Reaction–Diffusion Equation

This section is set aside to show that by using Galerkin in combination with the compactness methods, the solution of the two-dimensional semi-linear reaction–diffusion Equations (1)–(3) exits uniquely in the space

$$\begin{array}{c}\hfill L^{\infty }\left[(0,T);L^2(\Omega )\right]\cap L^2\left[(0,T);H_0^1(\Omega )\right]\cap L^4\left[(0,T);L^4(\Omega )\right]\mathrm{and}\frac{\mathsf{\partial }u}{\mathsf{\partial }t}\in L^2\left[(0,T);H^{-1}(\Omega )\right]\end{array}$$

(10)

for initial data given in (3). The a priori estimates obtained in the above analysis of the theoretical framework will be crucial to the NSFD-GM numerical scheme, which will be outlined in Section 4. In view of this, we proceed to show this by first stating the following variational or weak problem of the semi-linear reaction–diffusion Equations (1)–(3), finding $u(x,t)\in L^{\infty }\left[(0,T);L^2(\Omega )\right]\cap L^2\left[(0,T);H_0^1(\Omega )\right]\cap L^4\left[(0,T);L^4(\Omega )\right]$ such that

$$\begin{array}{c}\hfill 〈\frac{\mathsf{\partial }u}{\mathsf{\partial }t},v〉+〈\nabla u,\nabla v〉+〈\left(\right|u{|}^2-1)u,v〉=0\end{array}$$

(11)

$$\begin{array}{c}\hfill 〈u(x,0),v〉=〈u_0,v〉.\end{array}$$

(12)

for all $u_0,v\in H_0^1(\Omega )$. This is undertaken by making use of the orthonormal basis of $L^2$ functions denoted by $\left\{e_1,e_2,\cdots ,e_m\right\}\subset H_0^1\cap H^2(\Omega )$, where $m\in \mathbb{N}$. With these basic functions, we can make use of the test function v spanned by $v\in span\left\{e_1,e_2,\cdots ,e_m\right\}$ to denote and define the approximation of the solution u of the equation through

$$\begin{array}{c}\hfill u_m=\sum _{i=1}^m{\gamma }_i\left(t\right)e_i.\end{array}$$

(13)

We can then proceed to apply the above Galerkin framework (13) to the semi-linear reaction–diffusion Equations (1)–(3). The approximation $\left\{u_m\right\},m\in \mathbb{N}$ of the said equation satisfies the following Galerkin system of equations:

$$\begin{array}{c}\hfill \frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t}-\Delta u_m+P_m\left(\left(\right|u_m{|}^2-1)u_m\right)=0,\mathrm{on}\Omega \times (0,T)\end{array}$$

(14)

$$\begin{array}{c}\hfill u_m(x,t)=0\mathrm{on}\mathsf{\partial }\Omega \times (0,T)\end{array}$$

(15)

$$\begin{array}{c}\hfill u(x,0)=P_m{u}_0\mathrm{on}\Omega \end{array}$$

(16)

where the above projection $P_m$ denotes the orthogonal projection

$$\begin{array}{c}\hfill P_m:H^{-1}(\Omega )⟶{\mathcal{V}}_m\subset H^{-1}(\Omega ).\end{array}$$

(17)

and ${\mathcal{V}}_m$ is defined in Equation (9). This means that the operator is extended from $L^2(\Omega )$ into $H^{-1}(\Omega )$ and defined on the $H^{-1}(\Omega )$ by

$$\begin{array}{c}\hfill P_m\left(\sum _{k\in m}{\gamma }_m^k\left(t\right)u_k\right)=\sum _{k=1}^m{\gamma }_m^k\left(t\right)u_k.\end{array}$$

(18)

At this point, we should state that Equations (14)–(16) are also satisfied, with the discrete solution $u_m$ taking values in the finite dimensional subspace $V_m$. The connection between the semi-linear reaction–diffusion Equations (1)–(3) and their approximate Galerkin system of Equations (14)–(16) stated above validate the fact that these equations are equivalent, as seen classically in Temam, 1997 [11], and Evans, 1998 [32]. The above connection provides the framework to proceed and shows that the solution of the semi-linear reaction–diffusion equation exists uniquely. This can be achieved thanks to the next Theorem 1.

Theorem 1.

There exists a unique solution u of the semi-linear reaction–diffusion Equations (1)–(3) in the space given in (10), such that the variational solutions of Equations (11) and (12) are satisfied for the initial solution $u_0\in H_0^1(\Omega )$, with its Dirichlet boundary conditions.

We will use three subsections to prove this Theorem. These will be Section 3.1, where we will address the uniform approximation of the solution, followed by Section 3.2, where the boundedness of the approximate solution and passage to the limit will be achieved using the compactness method. The uniqueness of the solution of the problem will be shown in Section 3.3.

3.1. Uniform Approximate Solution Estimates of the Equation

To show the uniform approximation estimate of the semi-linear reaction–diffusion equation, we will first consider, here and after, that C will denote all constants independent of m. With this in mind, and $v\in H_0^1(\Omega )$ in Equations (11) and (12), the yield of the variational Equations (14)–(16) will be

$$\begin{array}{c}\hfill \frac{1}{2}\frac{d}{dt}{\parallel u_m\parallel }_{L^2}^2+{∥\nabla u_m∥}_{L^2}^2+{\int }_{\Omega }\left(|u_m{|}^2{u}_m-u_m\right)u_{m}dx=0\end{array}$$

(19)

where $v=u_m$ was set. Proceeding to use the Sobolev embedding Theorem and the fact that $L^4\subset H^1$, we bound the third term of (19) as follows:

$$\begin{array}{c}\hfill {\int }_{\Omega }{u}_m^2\left(\right|u_m{|}^2-1)dx\le \parallel u_m{\parallel }_{L^4}^4-{\parallel u_m\parallel }_{L^2}^2\end{array}$$

and introducing it back into Equation (19), yielded

$$\begin{array}{c}\hfill \frac{1}{2}\frac{d}{dt}\parallel u_m{\parallel }_{L^2}^2+{∥\nabla u_m∥}_{L^2}^2+\parallel u_m{\parallel }_{L^4}^4\le {\parallel u_m\parallel }_{L^2}^2.\end{array}$$

(20)

after taking $D=1$ for simplicity’s sake. Hence, taking the integral of both sides of (20) with respect to the time interval t yields

$$\begin{array}{c}\hfill \parallel u_m{\parallel }_{L^2}^2+{\int }_0^t\left[\parallel \nabla u_m{\parallel }_{L^2}^2+{\parallel u_m\left(s\right)\parallel }_{L^2}^4\right]ds\le \parallel u_0{\parallel }_{L^2}^2+{\int }_0^t{\parallel u_m\left(s\right)\parallel }_{L^2}^{2}ds.\end{array}$$

(21)

Applying the Gronwall inequalities, keeping only the term $\parallel u_m{\left(t\right)\parallel }_{L^2}^2$ on the left-hand side of (21), we obtain

$$\begin{array}{c}\hfill \parallel u_m{\parallel }_{L^2}^2\le C\left(\parallel u_0{\parallel }_{L^2}^2\right)e^T.\end{array}$$

(22)

Using (22) in (21), we find

$$\begin{array}{c}\hfill {\int }_0^t\left[\parallel \nabla u_m{\left(s\right)\parallel }_{L^2}^2+{\parallel u_m\left(s\right)\parallel }_{L^4}^4\right]ds\le C\left(\parallel u_0{\parallel }_{L^2}^2\right).\end{array}$$

(23)

Hence, in view of the inequalities (22) and (23), this shows that the apprimate solution $u_m$ is uniformly bounded in the space (10). We are now left to estimate the fact that ${∥\frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t}∥}_{H^{-1}}$ is also uniformly bounded in (11). To this end, we show, in view of (11),

$$\begin{array}{c}\hfill {\int }_0^T\left|〈\frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t},v〉\right|dx\le {\int }_0^T|\nabla u_m{|}_{L^2}^2,{|\nabla v|}_{L^2}^{2}dx+{\int }_0^T\left|〈u_m\left(\right|u_m{|}^2-1),v〉\right|dx\end{array}$$

(24)

We proceed with (28) to bound the first and second terms of the right-hand side of the inequality, to produce

$$\begin{array}{ccc}\hfill {\int }_0^T\left|〈\frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t},v〉\right|dx& \le & {\int }_0^T|\nabla u_m{|}_{L^2}^2,{|\nabla v|}_{L^2}^{2}dx+{\int }_0^T\left|〈u_m,v〉\right|dx+{\int }_0^T{\left|u_m\right|}_{L^4}^3{\left|v\right|}_{L^2}dx\hfill \\ & \le & {\int }_0^T|\nabla u_m{|}_{L^2}^2,{|\nabla v|}_{L^2}^{2}dx+{\int }_0^T{\left|u_m\right|}_{L^4}{\left|v\right|}_{L^2}dx+{\int }_0^T{\left|u_m\right|}_{L^4}^3{\left|v\right|}_{L^2}dx\hfill \end{array}$$

(25)

Using the inequalities (22) and (23), the Sobolev embedding Theorem $H^1\subset L^2$, the supremum over the time interval t and the Gagliardo Nirenberg inequality, the first, second and third terms on the right-hand side of (25) are bounded as follows:

$$\begin{array}{ccc}\hfill {\int }_0^T\left|〈\frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t},v〉\right|dx& \le & C_1+C_2\underset{0\le t\le T}{sup}{\parallel u\parallel }_{L^2}^{1/2}{\int }_0^T{\parallel \nabla u_m\parallel }_{L^2}^{1/2}\hfill \\ & +& C_3\underset{0\le t\le T}{sup}\parallel u_m{\parallel }_{L^2}^{3/2}{\int }_0^T{\parallel \nabla u_m\parallel }_{L^2}^{3/2}\hfill \\ & \le & C\hfill \end{array}$$

(26)

after using the fact that ${\parallel w\parallel }_{H^{-1}}={sup}_{v\in H_0^1}|<w,v>|$ and ${\parallel v\parallel }_{H^1}\le 1$, and hence we conclude from above that

$$\begin{array}{c}\hfill {\int }_0^T{\parallel \frac{\mathsf{\partial }{u}_m\left(s\right)}{\mathsf{\partial }t}\parallel }_{H^{-1}}ds\le C.\end{array}$$

(27)

All the analysis resulting in (22), (23) and (27), above, concludes that the sequence of solutions $\left\{u_m\right\},m\in \mathbb{N}$ is uniformly bounded in the following space (10).

3.2. Boundedness and Passage to the Limit of Approximate Solution of the Equation

In view of the analysis of the uniform boundedness of the approximate solution in Section 3.1, we proceed to show that the sequences of solution $\left\{u_m\right\}$ will converge strongly to the solution $u\left(t\right)$. To this end, we proceed by recalling the fact that we have obtained the following approximate solution $\left\{u_m\right\}$ defined on the $[0,T]$:

$$\begin{array}{c}\hfill u_m\mathrm{is} \mathrm{uniformly} \mathrm{bounded} \mathrm{in}{L}^{\infty }\left[(0,T);L^2(\Omega )\right]\\ \hfill u_m\mathrm{is} \mathrm{uniformly} \mathrm{bounded} \mathrm{in}{L}^2\left[(0,T);H_0^1(\Omega )\right]\\ \hfill u_m\mathrm{is} \mathrm{uniformly} \mathrm{bounded} \mathrm{in}{L}^4\left[(0,T);L^4(\Omega )\right]\\ \hfill \frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t}\mathrm{is} \mathrm{uniformly} \mathrm{bounded} \mathrm{in}{L}^2\left[(0,T);H^{-1}.(\Omega )\right]\end{array}$$

In view of the embedding of

$$\begin{array}{c}\hfill H_0^1(\Omega )\hookrightarrow L^2(\Omega )\hookrightarrow H^{-1}(\Omega )\end{array}$$

by Banach–Alaoglu’s Theorem found in [36], there exists a subsequence of $u_m$ still denoted by $u_m$, such that

$$\begin{array}{c}\hfill u_m⟶u\mathrm{weakly} \mathrm{star} \mathrm{in}{L}^{\infty }\left[(0,T);L^2(\Omega )\right]\\ \hfill u_m⟶u\mathrm{weakly} \mathrm{in}{L}^2\left[(0,T);H_0^1(\Omega )\right]\\ \hfill u_m⟶u\mathrm{weakly} \mathrm{in}{L}^4\left[(0,T);L^4(\Omega )\right]\\ \hfill \frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t}⟶\frac{\mathsf{\partial }u}{\mathsf{\partial }t}\mathrm{weakly} \mathrm{in}{L}^2\left[(0,T);H^{-1}.(\Omega )\right]\end{array}$$

and in view of the following Theorem 2 found in [36], $u_m⟶u\mathrm{strongly} \mathrm{in}{L}^2\left[(0,T);L^2(\Omega )\right].$

Theorem 2.

Suppose that $X\hookrightarrow Y\hookrightarrow Z$ are Banach spaces in which $X,Z$ are reflexive and X is compactly embedded in Y. Let $1<p<\infty$. If the functions $w_N:(0,T)⟶X$ are such that $\left\{w_N\right\}$ is uniformly bounded in $L^2\left[(0,T);X\right]$ and $\left\{\frac{\mathsf{\partial }u}{\mathsf{\partial }t}\right\}$ is uniformly bounded in $L^p\left[(0,T);Z\right]$, then there is a subsequence that converges strongly in $L^2\left[(0,T);Y\right]$.

With the strong convergence of the approximate solution $u_m$ in $L^2\left[(0,T);L^2(\Omega )\right]$, what remains to be shown now is that the solution satisfies Equation (12). To achieve this, we introduce another test function $\psi \left(t\right)$ with values $\psi \left(0\right)=1$ and $\psi \left(T\right)=0$ and also continuously differentiable on the interval $[0,T]$. With this in place, we proceed in view of Equation (19) and the function $\psi \left(t\right)$:

$$\begin{array}{c}\hfill 〈\frac{\mathsf{\partial }{u}_m}{\mathsf{\partial }t},v〉\psi \left(t\right)+〈\nabla u_m,\nabla v〉\psi \left(t\right)+〈u_m\left(\right|u_m{|}^2-1),v〉\psi \left(t\right)=0.\end{array}$$

(28)

If, in Equation (28), integration by parts with respect to t over the interval $[0,T]$ is applied, then we obtain

$$\begin{array}{ccc}\hfill -{\int }_0^T〈\frac{u_m}{\mathsf{\partial }t},\psi \left(t\right)〉vdt+{\int }_0^T〈\nabla u_m,\nabla v\psi \left(t\right)〉dt& +& {\int }_0^T〈u_m\left(\right|u_m{|}^2-1),v\psi \left(t\right)〉dt\hfill \\ & =& 〈u\left(0\right),v〉\psi \left(t\right).\hfill \end{array}$$

(29)

Using Theorem 2, the approximate solution $u_m$ is uniformly bounded. Then, passing to the limit, we have a yield in view of (29) as follows:

$$\begin{array}{ccc}\hfill -{\int }_0^T〈\frac{u}{\mathsf{\partial }t},\psi \left(t\right)〉vdt+{\int }_0^T〈\nabla u,\nabla v\psi \left(t\right)〉dt& +& {\int }_0^T〈u\left(\right|u{|}^2-1),v\psi \left(t\right)〉dt\hfill \\ & =& 〈u\left(0\right),v〉\psi \left(t\right).\hfill \end{array}$$

(30)

which holds in particular for $\psi \left(t\right)\in \mathcal{D}(0,T)$. This therefore means that the solution u in Equation (30) is satisfield in the distributional sense. Therefore, in view of Equations (28) and (30), this yields

$$\begin{array}{c}\hfill 〈u\left(0\right)-u_0,v〉=0\forall v\in H_0^1(\Omega )\end{array}$$

and Equation (12) is obtained as required.

3.3. Uniqueness of the Solution of the Equation

We devote this subsection to the uniqueness of the solution of the semi-linear reaction–diffusion Equations (1)–(3). We address this by letting $u_1$ and $u_2$ be two solutions of the equations, such that $u:=u_1-u_2$ and ${u|}_{\mathsf{\partial }\Omega }=0$. Since u satisfies Equations (1) and (2), then $u\left(0\right)=u_1\left(0\right)-u_2\left(0\right)=0.$ In view of this, we proceed, using Equation (1), to obtain

$$\begin{array}{c}\hfill \frac{\mathsf{\partial }u}{\mathsf{\partial }t}-\Delta u+u_1|u_1{|}^2-u_1-(u_2|u_2{|}^2-u_2)=0\end{array}$$

(31)

from which, after factorizing $|u_1{|}^2-|u_2{|}^2=\left(\right|u_1|+|u_2\left|\right)\left(\right|u_1|-|u_2\left|\right)$ and multiplying throughout by u, we obtain

$$\begin{array}{c}\hfill \frac{1}{2}{\parallel u\parallel }_{L^2}^2+{∥\nabla u∥}_{L^2}^2\le {\int }_{\Omega }{u\left|\right|}_{L^4}^2\left(|u_1{|}^2+1\right)dx+{\int }_{\Omega }{\left|u\right|}_{L^4}^2|u_2\left|\right|u_1|dx+{\int }_{\Omega }{\left|u\right|}_{L^4}^2{\left|u_2\right|}^{2}dx\end{array}$$

(32)

after using Hölder’s inequality. Estimating the right hand side of Inequality (32) thanks to the Gagliardo–Nirenberg inequality, we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left|u\right|}_{L^4}^2\left(|u_1{|}^2+|u_1\left|\right|u_2|+|u_2{|}^2+1\right)dx\le \frac{1}{2}{\parallel \nabla u\parallel }_{L^2}^2+{\parallel u\parallel }_{L^2}^2{\left({\left|u\right|}^2+|u_1\left|\right|u_2|+|u_2{|}^2+1\right)}^2\end{array}$$

(33)

after using the Young inequality. Introducing (33) back into (32) yields

$$\begin{array}{c}\hfill \frac{1}{2}{\parallel u\parallel }_{L^2}^2+\frac{1}{2}{∥\nabla u∥}_{L^2}^2\le C{\parallel u\parallel }_{L^2}^2\mathcal{Y}\end{array}$$

(34)

where $\mathcal{Y}={\left|u\right|}^2+|u_1\left|\right|u_2|+|u_2{|}^2+1$. Integrating the inequality (34) with respect to t over the interval $(0,T)$ and keeping only the term ${\parallel u\parallel }_{L^2}^2$ on the left-hand side yields

$$\begin{array}{c}\hfill {\parallel u\left(t\right)\parallel }_{L^2}^2\le C{\parallel u\left(0\right)\parallel }_{L^2}^2{e}^{{\int }_0^T{\mathcal{Y}}^{2}dt}=0,\forall t\ge 0\end{array}$$

after applying the Gronwall inequality. Hence, this proves the uniqueness of the solution of the problem.

4. The Design of the Coupled NSFD-GM Numerical Scheme

Instead of the theoretical analysis of the two-dimensional semi-linear reaction–diffusion equation in Section 3, we apply the a priori estimates obtained from the above section to analyze this section. In view of the complexity that comes from the analysis of the two-dimensional situation of semi-linear equations such as ours, we proceed to use the one-dimensional case in [20] as a reference model to analyze the stability and the convergence results from the designed NSFD-GM numerical scheme. In view of this, we have set aside this section to design and analyze the NSFD-GM numerical scheme mentioned in Section 1. This analysis will be achieved in two subsections, which are Section 4.1, where we will address the stability of the scheme and the optimal convergence of the scheme in both the $L^2$- as well as the $H^1$-norms, and Section 4.2, which will show that the scheme preserves the decaying properties of the exact solution. In view of the above three objectives, we start by stating the discrete version of the variational or weak forms (11) and (12), as follows: find $v_h:[0,T]⟶{\mathcal{V}}_h$, and the discrete solution such that

$$\begin{array}{c}\hfill 〈\frac{\mathsf{\partial }{u}_h}{\mathsf{\partial }t},v_h〉+〈\nabla u_h,\nabla v_h〉+〈\left(|u_h{|}^2-1\right)u_h,v_h〉=0\end{array}$$

(35)

$$\begin{array}{c}\hfill 〈u_h(x,0),v_h〉=〈P_h〉,\forall v_h\in {\mathcal{V}}_h\end{array}$$

(36)

where $P_h$ is the orthogonal projection onto ${\mathcal{V}}_h$.

The above discrete form connects us from the continuous to the discrete framework, which is geared toward the analysis of the numerical solution of (35) and (36). The above framework can proceed by assuming the regularity of the solution u of (11) and (12) and the subspace ${\mathcal{V}}_h\subset H_0^1(\Omega )$, as seen in [34]. Other useful inequalities to be used in this analysis will be

$$\begin{array}{c}\hfill \parallel P_{h}v-v\parallel \le C{h}^2{\parallel v\parallel }_{H^2},\mathrm{for}v\in H_0^1\cap H^2\end{array}$$

(37)

where $\parallel ·\parallel$ is the usual norm in $L^2$ and $H^1$ is a standard Sobolev space with some constant C. Also, if u is sufficiently smooth for a closed time interval $[0,T]$, then it is well-known in view of [37] that

$$\begin{array}{c}\hfill |u\left(t\right)-u_h\left(t\right)|\le C_1(u,C_2,C_3)h^2,\forall t\in [0,T]\end{array}$$

(38)

where $C_2$ is the bound on U and $\nabla u$ with $C_3$ as the constant in (38).

Setting in place the above desired framework, we continue and let the time step size $t_n=n\mathsf{\Delta }t$ for $n=0,1,2,\cdots ,\mathbb{N}$ over the time interval $[0,T]$. This then led us to find the NSFD-GM-approximate solution $\left\{U_h^n\right\}$ such that $U_h^n\approx u_h^n$ at each discrete time $t_n$ in the space ${\mathcal{V}}_h$ for sufficiently smooth functions. The stated approximation permits us to define the NSFD-GM scheme of the semi-linear reaction–diffusion equation as one which consists of a fully discrete solution $U_h^n\in {\mathcal{V}}_h$ such that for $v_h\in {\mathcal{V}}_h\subset H_0^1(\Omega )$,

$$\begin{array}{ccc}\hfill 〈{\delta }_n{U}_h^n\left(t\right),v_h〉+〈\nabla U_h^n,\nabla v_h〉+〈U_h^n\left(|U_h^n{|}^2-1\right),v_h〉& =& 0,\hfill \end{array}$$

(39)

$$\begin{array}{c}\hfill 〈U_h^n,v_h〉=〈P_h{u}_0,v_h〉,\end{array}$$

(40)

are satisfied, where

$$\begin{array}{c}\hfill {\delta }_n{U}_h^n=\frac{U_h^n-U_h^{n-1}}{\varphi \left(\mathsf{\Delta }t\right)}\mathrm{and}\varphi \left(\mathsf{\Delta }t\right)=\frac{e^{\lambda \mathsf{\Delta }t}-1}{\lambda }.\end{array}$$

(41)

The above new framework needs the following components:

(a)

The special and complicated function $\varphi \left(\mathsf{\Delta }t\right)$ where $\lambda >0$ is in such a way that

$$\begin{array}{c}\hfill 0<\varphi \left(\mathsf{\Delta }t\right)<1\mathrm{for}n=1,2,3,\cdots ,N\end{array}$$

(42)

(b)

If the nonlinear function $\left(|U_h^n{|}^2-1\right)U_h^n$ is made in such a way that its effect is negligible or even zero, then the scheme (39) will coincide with the exact scheme

$$\begin{array}{c}\hfill 〈\frac{U_h^{n+1}-U_h^n}{\varphi \left(\mathsf{\Delta }t\right)}〉+〈\nabla U_h^n,\nabla v_h〉=0\end{array}$$

(43)

which, in view of Michens [22], replicates or preserves the decaying-to-zero property of the scheme under investigation; this is actually the main point of the novelty of the method.

4.1. The Stability of the Coupled NSFD-GM Numerical Scheme

In this subsection, we show that the NSFD-GM scheme of (39) and (40) is stable. That is, we show that the numerical solution of the above scheme is uniformly bounded as the following Theorem 3:

Theorem 3.

If we assume that the semi-linear reaction–diffusion equation u in Equations (11) and (12) is regular, then for a given initial solution $U_h^0\in {\mathcal{V}}_h$, the numerical solution $U_h^n\left(t\right)$ of the scheme (39) and (40) will remain bounded and satisfy the following estimates:

$$\begin{array}{ccc}\hfill |U_h^0{|}^2& \le & |U_h^0{|}^2+2\varphi \left(\mathsf{\Delta }t\right)C(\Omega ),\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill \sum _{n=1}^N{|U^n-U_h^{n-1}|}^2& \le & |U_h^0{|}^2+2\varphi \left(\mathsf{\Delta }t\right)C(\Omega ).\hfill \end{array}$$

(45)

Proof. 

We prove the above theorem by first letting $v_h=U_h^n\left(t\right)$ in Equation (39) yield

$$\begin{array}{c}\hfill 〈U_h^n\left(t\right)-U_n^{n-1}\left(t\right),U_h^n\left(t\right)〉+\varphi \left(\mathsf{\Delta }t\right)\parallel \nabla U_h^n{\left(t\right)\parallel }_{L^2}^2+\varphi \left(\mathsf{\Delta }t\right)\parallel U_h^n{\left(t\right)\parallel }_{L^4}^4-\varphi \left(\mathsf{\Delta }t\right){\left|U_h^n\left(t\right)\right|}_{L^2}^2=0\end{array}$$

from where we have, in view of (20),

$$\begin{array}{c}\hfill 〈U_h^n\left(t\right)-U_n^{n-1}\left(t\right),U_h^n\left(t\right)〉+\varphi \left(\mathsf{\Delta }t\right)\parallel \nabla U_h^n{\left(t\right)\parallel }_{L^2}^2+\varphi \left(\mathsf{\Delta }t\right)\parallel U_h^n{\left(t\right)\parallel }_{L^4}^4\le \varphi \left(\mathsf{\Delta }t\right){\parallel U_0^1\left(t\right)\parallel }_{L^2}^2\end{array}$$

(46)

It is well-known, in view of the first term of the left-hand side of the inequality (58), that

$$\begin{array}{c}\hfill 〈U_h^n\left(t\right)-U_h^{n-1}\left(t\right),U_h^n\left(t\right)〉=\frac{1}{2}|U_h^n{|}^2-\frac{1}{2}|U_h^{n-1}{|}^2+\frac{1}{2}{|U_h^n-U_h^{n-1}|}^2.\end{array}$$

and re-introducing this equality back into (58) yields

$$\begin{array}{ccc}\hfill |U_h^n{|}^2-|U_h^{n-1}{|}^2+{|U_h^n-U_h^{n-1}|}^2& +& 2\varphi \left(\mathsf{\Delta }t\right){∥\nabla U_h^n∥}_{L^2}^2+2\varphi \left(\mathsf{\Delta }t\right){∥U_h^n∥}_{L^4}^4\hfill \\ & \le & 2\varphi \left(\mathsf{\Delta }t\right){∥U_h^n∥}_{L^2}^2.\hfill \end{array}$$

(47)

Taking the sum to $\mathbb{N}$ of the above inequality (47), we obtain

$$\begin{array}{ccc}\hfill |U_h^n{|}^2+\sum _{n=1}^{\mathbb{N}}{|U_h^n-U_h^{n-1}|}^2& +& 2\varphi \left(\mathsf{\Delta }t\right)\sum _{n=1}^{\mathbb{N}}{∥\nabla U_h^n∥}_{L^2}^2+2\varphi \left(\mathsf{\Delta }t\right)\sum _{n=1}^{\mathbb{N}}{∥U_h^n∥}_{L^4}^4\hfill \\ & \le & 2\varphi \left(\mathsf{\Delta }t\right)\sum _{n=1}^{\mathbb{N}}{∥U_h^n∥}_{L^2}^2+{\left|U_h^0\right|}^2.\hfill \end{array}$$

(48)

In view of (22) and (23), we can immediately read the results (44) and (45) from (48) as required. □

4.2. Optimal Convergence of the Coupled NSFD-GM Numerical Scheme

We devote this subsection to showing that the numerical solution obtained from the NSFD-GM scheme converges optimally in both $L^2$- and $H^1$-norms, and also that these solutions replicate or preserve the decay-to-zero properties of the exact solution. To achieve these two objectives, we first state without proof the following results from Shen [38].

Lemma 1.

Let $\mathsf{\Delta }t,\gamma$ and $a_k,b_k,d_k,{\gamma }_k$ for the integer $k\ge 0$ be non-negative numbers such that

$$\begin{array}{c}\hfill a_J+\sum _{k=0}^J{b}_k\mathsf{\Delta }t\le \sum _{k=0}^J{d}_k{a}_J\mathsf{\Delta }t+\sum _{k=0}^J{\gamma }_k\mathsf{\Delta }t+\gamma ,\forall J\ge 0.\end{array}$$

(49)

Suppose that

$$\begin{array}{c}\hfill d_k\mathsf{\Delta }t<1and set{\sigma }_k={(1-d_k\mathsf{\Delta }t)}^{-1},\forall k\ge 0.\end{array}$$

(50)

Then, we have

$$\begin{array}{c}\hfill a_J+\sum _{k=0}^J{b}_k\mathsf{\Delta }t\le exp\left(\sum _{k=0}^J{d}_k\mathsf{\Delta }t\right)\left(\sum _{k=0}^J{\gamma }_k\mathsf{\Delta }t+\gamma \right)\forall J\ge 0.\end{array}$$

(51)

With the above Lemma 1 and NSFD-GM framework in mind, we can then state and prove the error estimate in Theorem 4, next.

Theorem 4.

We assume that ${\mathsf{\Phi }}_k$ is a non-negative number and that the continuous and discrete solutions of the semi-linear reaction–diffusion Equations (39) and (40), respectively, exist uniquely and satisfy

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_k\varphi \left(\mathsf{\Delta }t\right)<1and{\sigma }_k={\left(1-{\mathsf{\Phi }}_k\varphi \left(\mathsf{\Delta }t\right)\right)}^{-1},\forall k\ge 0.\end{array}$$

Then, the following error estimate

$$\begin{array}{c}\hfill \parallel u\left(t_J\right)-U_h\left(t_J\right)\parallel +\varphi \left(\mathsf{\Delta }t\right)\sum _{k=0}^J{\left|\nabla (u\left(t_J\right)-U_h\left(t_J\right))\right|}^2\le C\left(t_J\right){\left(\varphi \left(\mathsf{\Delta }t\right)\right)}^2,\forall J\ge 0.\end{array}$$

(52)

is satisfied.

Proof. 

The above Theorem is proved by using the implicit non-standard finite difference method in the time as follows:

$$\begin{array}{c}\hfill \frac{U_{n+1}-U_n}{\varphi \left(\mathsf{\Delta }t\right)}=\mathsf{\Delta }{U}_{n+1}+U_{n+1}\left(1-|U_{n+1}{|}^2\right).\end{array}$$

(53)

In view of the non-standard Taylor’s integral Theorem, we find

$$\begin{array}{ccc}\hfill \frac{u(t_{n+1}-u\left(t_n\right))}{\varphi \left(\mathsf{\Delta }t\right)}& =& \frac{\mathsf{\partial }u\left(t_{n+1}\right)}{\mathsf{\partial }t}-\frac{1}{2}{\int }_{t_n}^{t_{n+1}}\frac{{\mathsf{\partial }}^{2}u\left(t\right)}{\mathsf{\partial }{t}^2}(t_{n+1}-t)dt\hfill \\ & =& \mathsf{\Delta }u\left(t_{n+1}\right)+u\left(t_{n+1}\right)\left(1-|u\left(t_{n+1}\right){|}^2\right)\hfill \\ & -& \frac{1}{2}{\int }_{t_n}^{t_{n+1}}\frac{{\mathsf{\partial }}^{2}u\left(t\right)}{\mathsf{\partial }{t}^2}(t_{n+1}-t)dt.\hfill \end{array}$$

(54)

Subtracting Equation (54) from (53) and noting that ${\mathsf{\Theta }}_n=u\left(t_n\right)-U_n$ yields

$$\begin{array}{ccc}\hfill \frac{1}{\varphi \left(\mathsf{\Delta }t\right)}\left[{\mathsf{\Theta }}_{n+1}-{\mathsf{\Theta }}_n,{\mathsf{\Theta }}_{n+1}\right]& =& 〈u^{n+1}\left(1-|u^{n+1}{|}^2\right)-U_{n+1}\left(1-|U_{n+1}{|}^2\right),{\mathsf{\Theta }}_{n+1}〉\hfill \\ & -& \parallel \nabla {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2+\frac{1}{2}{\int }_{t_n}^{t_{n+1}}〈\frac{{\mathsf{\partial }}^{2}u\left(t\right)}{\mathsf{\partial }{t}^2},{\mathsf{\Theta }}_{n+1}〉(t-t_{n+1})dt\hfill \end{array}$$

(55)

after setting $u^{n+1}=u\left(t_{n+1}\right)$ and multiplying Equation (53) by ${\mathsf{\Theta }}_{n+1}$. Estimating the first term of the right-hand side of Equation (55), we find

$$\begin{array}{ccc}\hfill {\int }_{\Omega }\left|〈\left(u^{n+1}\left(1-|u^{n+1}{|}^2\right)-U_{n+1}\left(1-|U_{n+1}{|}^2\right)\right),{\mathsf{\Theta }}_{n+1}〉\right|dx& \le & {\int }_{\Omega }{\left|{\mathsf{\Theta }}_{n+1}\right|}_{L^4}^2{\left|\left(1-|u^{n+1}{|}^2\right)\right|}^{2}dx\hfill \\ & +& {\int }_{\Omega }|{\mathsf{\Theta }}_{n+1}{|}_{L^4}^2{\left|U_{n+1}\right|}_{H^1}^{2}dx\hfill \\ & +& {\int }_{\Omega }|{\mathsf{\Theta }}_{n+1}{|}_{L^4}^2|U_{n+2}{|}_{H^1}\left|u^{n+1}\right|dx\hfill \end{array}$$

(56)

after using the fact that $|U_{n+1}{|}^2-|u^{n+1}{|}^2=\left(\right|U_{n+1}|-|u^{n+1}\left|\right)\left(\right|U_{n+1}|+|u^{n+1}\left|\right)$. Estimating each term on the right-hand side of the inequality (56) using Gagliardo–Nirenberg and Young’s inequality with the fact that $H^1\subset L^{\infty }$ yields the following inequality, beginning with the third term:

$$\begin{array}{c}\hfill {\int }_{\Omega }|{\mathsf{\Theta }}_{n+1}{|}_{L^4}^2|U_{n+1}||u^{n+1}|dx\le \frac{ϵ}{2}\parallel \nabla {\mathsf{\Theta }}_{n+1}{\parallel }^2+\frac{1}{2ϵ}\parallel {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2\parallel U_{n+1}{\parallel }_{L^{\infty }}^2{\parallel u^{n+1}\parallel }_{L^2}^2.\end{array}$$

(57)

This is followed by the second term, as follows,

$$\begin{array}{c}\hfill {\int }_{\Omega }|{\mathsf{\Theta }}_{n+1}{|}_{L^4}^2|U_{n+1}{|}^{2}dx\le \frac{ϵ}{2}\parallel \nabla {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2+\frac{1}{2ϵ}\parallel {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2{\parallel U_{n+1}\parallel }_{L^{\infty }}^2\end{array}$$

(58)

and lastly by the first term:

$$\begin{array}{c}\hfill {\int }_{\Omega }|{\mathsf{\Theta }}_{n+1}{|}_{L^4}^2{\left|\left(1-|u^{n+1}{|}^2\right)\right|}^{2}dx\le \frac{ϵ}{2}\parallel \nabla {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2+\frac{1}{2ϵ}{\parallel {\mathsf{\Theta }}_{n+1}\parallel }_{L^2}^2{\left|\left(1-|u^{n+1}{|}^2\right)\right|}^2\end{array}$$

(59)

Re-introducing (57)–(59) into inequality (56) yields

$$\begin{array}{ccc}\hfill & & {\int }_{\Omega }\left|〈\left(u^{n+1}\left(1-|u^{n+1}{|}^2\right)-U_{n+1}\left(1-|U_{n+1}{|}^2\right)\right),{\mathsf{\Theta }}_{n+1}〉\right|dx\hfill \\ & \le & \frac{3ϵ}{2}\parallel \nabla {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2+\frac{3}{2ϵ}{\parallel {\mathsf{\Theta }}_{n+1}\parallel }_{L^2}^2\left[|u^{n+1}{|}_{H^1}^2+|U_{n+1}{|}_{H^1}^2+|U_{n+1}{|}_{H^1}^2{|u^{n+1}|}_{H^1}^2+1\right]\hfill \end{array}$$

(60)

By estimating the third term of the right-hand side of the inequality (55), we find

$$\begin{array}{c}\hfill \left|\frac{1}{2\varphi \left(\mathsf{\Delta }t\right)}{\int }_{t_n}^{t_{n+1}}〈\frac{{\mathsf{\partial }}^{2}u\left(t\right)}{\mathsf{\partial }{t}^2},{\mathsf{\Theta }}_{n+1}〉(t-t_{n+1})dt\right|\le \frac{C}{2\varphi \left(\mathsf{\Delta }t\right)}|\nabla {\mathsf{\Theta }}_{n+1}|{\int }_{t_n}^{t_{n+1}}\left|\frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{t}^2}\right||t-t_{n+1}|dt.\end{array}$$

(61)

because according to Poincare inequality, $|{\mathsf{\Theta }}_{n+1}{|}_{H_0^1}\le C{|\nabla {\mathsf{\Theta }}_{n+1}|}_{L^2}$. Using the fact that there exists a function

$$\begin{array}{c}\hfill \varphi \left(\mathsf{\Delta }{t}_n\right)<\varphi \left(\mathsf{\Delta }t\right)<\varphi \left(\mathsf{\Delta }{t}_{n+1}\right)\mathrm{for}{t}_n<t<t_{n+1}\end{array}$$

yields

$$\begin{array}{c}\hfill {\left({\int }_{t_n}^{t_{n+1}}|t-t_{n+1}|\right)}^{1/2}\le \varphi \left(\mathsf{\Delta }t\right){(t-t_{n+1})}^{1/2}\le {\left(\varphi \left(\mathsf{\Delta }t\right)\right)}^{1/2}.\end{array}$$

Introducing this and Hölder’s inequality in (61), we obtain

$$\begin{array}{c}\hfill \left|\frac{1}{2\varphi \left(\mathsf{\Delta }t\right)}〈\frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{t}^2},{\mathsf{\Theta }}_{n+1}〉(t-t_{n+1})dt\right|\le \frac{ϵ}{2}\nabla {\mathsf{\Theta }}_{n+1}{|\parallel }_{L^2}^2+\frac{C}{2ϵ}\varphi \left(\mathsf{\Delta }t\right){\int }_{t_n}^{t_{n+1}}{\left|\frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}\right|}_{H^{-1}}^{2}dt,\end{array}$$

(62)

after using Young’s inequality for arbitrary $ϵ>0$. Re-introducing the inequalities (60) and (62) into (55) produces the following for $ϵ=1/2$:

$$\begin{array}{c}\hfill \frac{1}{\varphi \left(\mathsf{\Delta }t\right)}\left[{\mathsf{\Theta }}_{n+1}-{\mathsf{\Theta }}_n,{\mathsf{\Theta }}_{n+1}\right]+\frac{1}{2}\parallel \nabla {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2\le C{\parallel {\mathsf{\Theta }}_{n+1}\parallel }_{L^2}^2{\Psi }_{n+1}+C{\mathsf{\Phi }}_{n+1}\varphi \left(\mathsf{\Delta }t\right)\end{array}$$

(63)

where

$$\begin{array}{c}\hfill {\Psi }_{n+1}=|U_{n+1}{|}_{H^1}^2+|U_{n+1}{|}_{H^1}^2|u^{n+1}{|}_{H^1}^2+{||}_{H{H}^1}^2+1\end{array}$$

and

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_{n+1}={\int }_{t_n}^{t_{n+1}}{\left|\frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{t}^2}\right|}_{H^{-1}}^{2}dt.\end{array}$$

In view of inequality (63), it is well known that the first term of the left-hand side is

$$\begin{array}{c}\hfill \left({\mathsf{\Theta }}_{n+1}-{\mathsf{\Theta }}_n,{\mathsf{\Theta }}_{n+1}\right)=\frac{1}{2}\left[|{\mathsf{\Theta }}_{n+1}{|}_{L^2}^2-|{\mathsf{\Theta }}_n{|}_{L^2}^2+{|{\mathsf{\Theta }}_{n+1}+{\mathsf{\Theta }}_n|}_{L^2}^2\right]\end{array}$$

and introducing this back into inequality (63),

$$\begin{array}{c}\hfill |{\mathsf{\Theta }}_{n+1}{|}_{L^2}^2-|{\mathsf{\Theta }}_n{|}_{L^2}^2+|{\mathsf{\Theta }}_{n+1}-{\mathsf{\Theta }}_n{|}_{L^2}^2+\varphi \left(\mathsf{\Delta }t\right)\parallel \nabla {\mathsf{\Theta }}_{n+1}{\parallel }_{L^2}^2\le C\varphi \left(\mathsf{\Delta }t\right){\parallel {\mathsf{\Theta }}_{n+1}\parallel }_{L^2}^2{\Psi }_{n+1}+C\varphi {\left(\mathsf{\Delta }t\right)}^2{\mathsf{\Phi }}_{n+1}.\end{array}$$

(64)

Arranging the terms in inequality (64) and setting $a_k={\parallel {\mathsf{\Theta }}_{n+1}\parallel }_{L^2}^2$ and $b_k={\parallel \nabla {\mathsf{\Theta }}_{n+1}\parallel }_{L^2}^2$ and summing partially for $k=0,1,2,\cdots ,n-1$ and also using the fact that $a_0={\mathsf{\Theta }}_0=u_0-U_0=0$, we obtain

$$\begin{array}{c}\hfill a_n+\sum _{k=0}^n{b}_k\varphi \left(\mathsf{\Delta }t\right)\le \sum _{k=0}^n{a}_k\varphi \left(\mathsf{\Delta }t\right){\Psi }_{n+1}+\sum _{k=0}^n\varphi {\left(\mathsf{\Delta }t\right)}^2{\mathsf{\Phi }}_{n+1}.\end{array}$$

(65)

Applying Lemma 1 in inequality (65) yields

$$\begin{array}{c}\hfill a_n+\sum _{k=0}^n{b}_k\varphi \left(\mathsf{\Delta }t\right)\le exp\left(\sum _{k=0}^n{\sigma }_k\varphi \left(\mathsf{\Delta }t\right){\Psi }_{n+1}\right)\left(\sum _{k=0}^n{\mathsf{\Phi }}_{n+1}{\left(\varphi \left(\mathsf{\Delta }t\right)\right)}^2\right)\end{array}$$

(66)

provided that ${\Psi }_{n+1}\varphi \left(\mathsf{\Delta }t\right)<1$ and ${\sigma }_k={\left(1-{\Psi }_{n+1}\varphi \left(\mathsf{\Delta }t\right)\right)}^{-1}\forall k\ge 0.$ Since $a_n,b_k,{\Psi }_{n+1}$ and ${\mathsf{\Phi }}_{n+1}$ are all positive series, then in view of Lemma 1,

$$\begin{array}{c}\hfill a_n+\sum _{k=0}^n{b}_k\varphi \left(\mathsf{\Delta }t\right)\le C{\left(\varphi \left(\mathsf{\Delta }t\right)\right)}^2\end{array}$$

(67)

and the proof of the theorem is completed. □

The error estimate shown above leads to the following optimal rate of convergence in both the $L^2$- and $H^1$-norms, as follows.

Theorem 5.

Assuming that the above error estimate in Theorem 4 is satisfied, then the numerical solution of the semi-linear reaction–diffusion scheme (39) and (40) has the following optimal rate of convergence:

$$\begin{array}{c}\hfill \parallel u\left(t\right)-U_h{\left(t\right)\parallel }_{L^2}\le C\left(t\right)\left(h^2+\varphi \left(\mathsf{\Delta }t\right)\right)\end{array}$$

using the NSFD-GM method, where $C\left(t\right)$ depends on t. Furthermore, the discrete solution $U_h\left(t\right)$ replicates all the qualitative properties of the exact solution of the equation under investigation.

Proof. 

We proceed to prove the above theorem by using the following error decomposition equation:

$$\begin{array}{ccc}\hfill \parallel u\left(t\right)-U_h{\left(t\right)\parallel }_{L^2}& =& \parallel u\left(t\right)-P_{h}u\left(t\right)+P_{h}u\left(t\right)-U_h{\left(t\right)\parallel }_{L^2}\hfill \\ & \le & \parallel {\xi }_n{\parallel }_{L^2}+{\parallel {\eta }_n\parallel }_{L^2}\hfill \end{array}$$

(68)

where $\parallel {\xi }_n{\parallel }_{L^2}={\parallel u\left(t\right)-P_{h}u\left(t\right)\parallel }_{L^2}$ and $\parallel {\eta }_n{\parallel }_{L^2}={\parallel P_{h}u\left(t\right)-U_h\left(t\right)\parallel }_{L^2}$. In view of the above inequality (68), the estimate $\parallel {\xi }_n{\parallel }_{L^2}$ represents the error inherent in the Galerkin approximation of the linearized reaction–diffusion equation, and that of the estimate $\parallel {\eta }_n{\parallel }_{L^2}$ is the error caused by the nonlinearity in the problem. The above error decomposition in Equation (68) further leads to the following inequality:

$$\begin{array}{c}\hfill \parallel u\left(t\right)-U_h{\left(t\right)\parallel }_{L^2}\le C\left(t_n\right)h^2+C\left(t_n\right){\left(\varphi \left(\mathsf{\Delta }t\right)\right)}^2,\forall t\in [t_n,t_{n+1}]\end{array}$$

(69)

after using inequalities (38) and (67). With these inequalities in place, we conclude without difficulty that inequality (69) is indeed optimal.

To complete the preserving of the qualitative properties of the exact solution, we proceed to show, in view of the remark from Mickens [22], that the above scheme was designed for

$$\begin{array}{c}\hfill \varphi \left(\mathsf{\Delta }t\right)=\frac{e^{\lambda \mathsf{\Delta }t}-1}{\lambda }\approx \mathsf{\Delta }t+O\left({\left(\mathsf{\Delta }t\right)}^2\right).\end{array}$$

(70)

Based on the above approximation (70), we observe that as $\mathsf{\Delta }t⟶0$, the function $\varphi \left(\mathsf{\Delta }t\right)\approx \mathsf{\Delta }t$. In view of this, we deduced that the numerical schemes (39) and (40) converge point-wise in ${\mathcal{V}}_h\subset H_0^1(\Omega )$ to the solution u as $\mathsf{\Delta }t⟶0$ according to the compactness Theorem. We justify this as follows: If we choose the source term of our scheme (39) to be $U_h^0\in H_0^1(\Omega )$ and $\mathbf{F}\in L^2\left[(0,T);H^{-1}(\Omega )\right]$, then we have

$$\begin{array}{c}\hfill 〈{\delta }_n{U}_h^n\left(t\right),v_h〉+〈\nabla U_h^n,\nabla v_h〉+〈U_h^n\left(|U_h^n{|}^2-1\right),v_h〉=\mathbf{F}.\end{array}$$

(71)

If we, in addition, let the support of $\mathbf{F}$ be very small, so much so that the test function $v_h=1$ far inside the support, say ${\Omega }_1\subset \Omega$ and F, is regular, then integrating Equation (71) over $\Omega$ will culminate in the fact that the solution over $\Omega$ is equivalent to the point-wise convergence of the scheme (71). In view of [31], the assertion that the scheme NSFD-GM replicates the qualitative properties of the exact scheme (43) is achieved. For more on this, see [20,28,29,30]. This completes the second part of the proof. □

5. Numerical Experiments

Numerical experiments will be performed in this section to compute the numerical solution of the nonlinear reaction–diffusion Equations (1)–(3). These experiments will be computed over a two-dimensional domain $\Omega =[0,1]\times [0,1]$ and the time interval $[0,T]$. The domain $\Omega$ will be regularly discretized into uniform triangular meshes denoted by ${\mathcal{J}}_h$, where h is the mesh size and the basic functions used will be linear Lagrange-type functions. The mesh size will be defined by $h=\frac{1}{M}$, where M is the number of nodes in the triangulations. The time space $[0,T]$ will be discretized with a step size of $\mathsf{\Delta }t$. The right-hand side of Equations (1)–(3), denoted by $f(x,t)$, will be determined by introducing a carefully chosen example:

$$\begin{array}{c}\hfill u(x_1,x_2,t)=(1+2t^2)cos\left(2\pi x_1^2\right)cos\left(2\pi x_2^2\right).\end{array}$$

(72)

We will proceed to use $f(x,t)$ in the NSFD-GM scheme of the semi-linear reaction–diffusion Equation (39) to give the following numerical computable scheme:

$$\begin{array}{c}\hfill (M+\varphi \left(\mathsf{\Delta }t\right)A)-\varphi \left(\mathsf{\Delta }t\right)M=f(x,t)+(M+\varphi \left(\mathsf{\Delta }t\right)A)+M{u}_0-\varphi \left(\mathsf{\Delta }t\right)F\left(u\right)\end{array}$$

(73)

where

(a)

M is the mass matrix;

(b)

A is the stiffness matrix;

(c)

$M_F$ is the mass matrix with respect to the semi-linear function $F\left(u\right)$;

(d)

$f(x,t)$ is the prescribed right hand side of the Equation (39).

In view of the above scheme (73), $(M+\varphi (\mathsf{\Delta }t\left)A\right)-\varphi \left(\mathsf{\Delta }t\right)M$ is the Jacobian matrix and $(M+\varphi \left(\mathsf{\Delta }t\right)A)+M{u}^{n-1}-\varphi \left(\mathsf{\Delta }t\right)F\left(u\right)$ is the Residual vector. With the above scheme, we proceed by evaluating the matrices $M,A$ and $M_F$ locally over a reference element and the basic function $v_h$. Using this, together with Newton’s iterative method using the following initial solution (74), the numerical solution from the scheme can be obtained:

$$\begin{array}{c}\hfill u_0=u(x_1,x_2,0)=cos\left(2\pi x_1^2\right)cos\left(2\pi x_2^2\right).\end{array}$$

(74)

The above experiments were conducted using the software Matblab 7.100(R2014a). The following specifications were used: $\lambda =4$, denoting the value of the parameter on the complicated denominator function $\varphi \left(\mathsf{\Delta }t\right)$; the final time $T=1.0$ and $\mathsf{\Delta }t=0.01$; and M ranging from $40,80,120,160,200$, and 240 to 280. With the above data, the Newton’s iterative method will yield the results in Figure 1, Figure 2 and Figure 3.

After the figures follow the tabular illustrations of the error and rate of convergence in the solutions, shown in

Table 1. NSFD-GM error in $L^2$-norm and error in $H^1$-norm.

M	Error in ${\mathit{L}}^2$-Norm	Rate of ${\mathit{L}}^2$	Error in ${\mathit{H}}^1$-Norm	Rate of ${\mathit{H}}^1$
40	3.0412 × ${10}^3$		2.1902 × ${10}^1$
80	1.2437 × ${10}^2$	1.29	1.4550 × ${10}^1$	0.59
120	7.2236 × ${10}^3$	1.34	1.0955 × ${10}^1$	0.70
160	4.5851 × ${10}^3$	1.58	8.6032 × ${10}^2$	0.84
200	3.1517 × ${10}^3$	1.68	7.0851 × ${10}^2$	0.87
240	2.2741 × ${10}^3$	1.79	6.0239 × ${10}^2$	0.89
280	1.6993 × ${10}^3$	1.89	5.2113 × ${10}^2$	0.94

and

Table 2. SFD-GM error in $L^2$-norm and error in $H^1$-norm.

M	Error in ${\mathit{L}}^2$-Norm	Rate of ${\mathit{L}}^2$	Error in ${\mathit{H}}^1$-Norm	Rate of ${\mathit{H}}^1$
40	3.0103 × ${10}^2$		2.0301 × ${10}^1$
80	1.2311 × ${10}^2$	1.29	1.3209 × ${10}^1$	0.62
120	7.0352 × ${10}^3$	1.38	9.6276 × ${10}^2$	0.78
160	4.4784 × ${10}^3$	1.57	7.5826 × ${10}^2$	0.83
200	3.0715 × ${10}^3$	1.69	6.2586 × ${10}^2$	0.86
240	2.2203 × ${10}^3$	1.78	5.3309 × ${10}^2$	0.88
280	1.6643 × ${10}^3$	1.89	4.6331 × ${10}^2$	0.91

for a specified value of $T=1.0$.

Observations 1.

Our expectations were that the rate of convergence in the $L^2$-norm would be 2 and that in the $H^1$-norm it would be 1, using both the NSFD-GM and SFD-GM schemes. To our great surprise, the results tended to be approximately 2 and 1 for both the $L^2$- and $H^1$-norms, respectively. Even though these values were not far from their expected values, the values of the NSFD-GM scheme in both norms were larger than those from their given counterpart SFD-GM scheme. This advantage of the NSFD-GM scheme over the SFD-GM scheme might be because of the principle of its design, keeping aside its effectiveness, accuracy and viability. It is also well-known and shown in the proof of Theorem 5, above, that the numerical solution from the NSFD-GM scheme exhibits the preservation or replication of the qualitative properties of the exact solution of the problem under investigation. These two reasons are what make the study very interesting, and the properties convince one to consider the NSFD-GM scheme to be a favorite in terms of a fair alternative over a more traditional SFD-GM scheme. The major advantage of the NSFD-GM over the more traditional SFD-GM scheme could be drawn from their rates of convergence, found in the two tables above. These rates of convergence values or results are clear and speak for themselves.

6. Conclusions and Future Remarks

The design of a reliable scheme consisting of the nonstandard finite difference method in time and the Galerkin combined with the compactness method in the space variables to analyze the semi-linear reaction–diffusion equation is presented in this paper. The analysis of the scheme began with the use of the Galerkin in combination with the compactness method to show that the weak solution of the problem existed uniquely in the space (10). We proceeded using a priori estimates from the existence of the solution process to show that the numerical scheme was stable and converged optimally in the $L^2$- and $H^1$-norms. Additionally, it was also shown that the numerical scheme preserves the decaying properties of the exact solution. Finally, with a prescribed example, numerical experiments were presented to validate the proposed theory. The results obtained from the study speak for themselves, for they were accurate and viable, as expected. In view of the results, the designed scheme could be seen as a fair and a justified alternative to some of the traditional coupling schemes, consisting of the finite difference method in the time, in combination with either the Galerkin or the finite element or even another finite difference method in the space variable itself.

In future, we intend to extend the technique to systems of semi- or quasi-linear parabolic problems such as the ones in [39,40] that have meaning in real life; in addition to this, we would also like to carry out some comparison studies with some similar schemes. We can also try out the technique with hyperbolic problems, as well. Another form of interesting study we would like to conduct is a study of some semi- or quasi-linear problems over non-smooth geometry using weighted Sobolev spaces.