1 Introduction

Singularly perturbed problems (SPPs) arise in the mathematical modeling of several practical problems in engineering and applied mathematics, for example, in describing the theory of gyroscopes [25], in studying linear spring-mass system without damping but with forcing and a small spring constant [31], in variational problems in control theory [10], etc., where the higher order derivative appears multiplied by a small parameter. The fascinating aspect of SPPs is that their solutions have boundary and/or interior layers (the regions in which there are steep gradients). Because of this reason, classical numerical schemes are not appropriate to accurately and efficiently solve these problems. This has led to the development of special numerical approaches for SPPs. The developed numerical methods for SPPs are called as parameter-robust or parameter-uniform or uniformly convergent, meaning that the convergence of the method is independent of the involved parameter. They are developed considering fitted operators, fitted meshes, and domain decomposition approaches [1, 2, 9, 11, 14,15,16, 26, 28,29,30,31, 35]. Further, some recent advancements in finite difference methods can be seen in [3,4,5,6].

Consider the following SPP

$$\begin{aligned} \begin{array}{rr} Lu:=u_t(x,t)-\varepsilon u_{xx}(x,t)+a(x,t)u(x,t) =f(x,t),~~ (x,t)\in \Omega :={\textsf{D}}\times (0,T], \end{array} \end{aligned}$$
(1)

with the Robin boundary conditions

$$\begin{aligned}{} & {} u(0,t)-\sqrt{\varepsilon } u_{x}(0,t)=g_\ell (t),~u(1,t)+\sqrt{\varepsilon } u_{x}(1,t)=g_r(t),~t\in (0,T], \end{aligned}$$
(2)
$$\begin{aligned}{} & {} \left. \begin{array}{ll} u(x,0)=g_b(x),~ x\in \overline{{\textsf{D}}}, \end{array} \right. \end{aligned}$$
(3)

where \({\textsf{D}}=(0,1),\) \( a(x,t)\ge \alpha >0 \) on \({\overline{\Omega }},\) and \(~0<\varepsilon \le 1\) is known as the perturbation parameter. It is known that problem (1)–(3) has a unique solution exhibiting boundary layers near \(x=0\) and \(x=1\) [12, 22, 37]. Such problems arise in the modelling of certain types of bioswitches [41].

Singularly perturbed problems similar to (1)–(3) with Dirichlet-type boundary conditions have been studied extensively in the literature (see [7, 18, 20, 24, 27, 34, 38] and the references therein). However, there are only a very few studies of such problems with Robin boundary conditions (RBCs) [12, 13, 22, 37]. Note that all these studies are based on the fitted mesh approach.

Domain decomposition is shown to be a very versatile and effective approach for solving problems in ordinary and partial differential equations [8, 17, 32]. The original approach dates back to 19th century [36]. However, the approach flourished more in the last three decades. Schwarz waveform relaxation (SWR) is a special approach that is very popular for solving time dependent problems [32]. In this technique, the space-time domain is split into several subdomains. Then, on each subdomain across the whole time interval the solution is computed and the exchange of time-space boundary values takes place. We can treat the subdomains differently using this approach. Also, we have the flexibility to locally address any singularity in the solution. Further, a non uniform mesh can be avoided with this approach. Lastly, one can speed-up the computations by implementing it on parallel computers. Domain decomposition methods for SPPs have been developed in [19,20,21, 33, 34, 39, 40] and the references therein. Note that in all these works model problems with Dirichlet boundary conditions are considered. Moreover, we are not aware of any studies involving the domain decomposition approach for SPPs with Robin boundary conditions.

Hence, the objective of this paper is two-fold: first, to introduce a domain decomposition method of SWR type to numerically solve problem (1)–(3), and second, to present a parameter-uniform convergence analysis of the introduced method. We consider a space-time partitioning of the original domain using the Shishkin transition point. The PDE associated with problem (1)–(3) in each subdomain is discretized by the finite difference scheme, while the Robin boundary conditions of problem (1)–(3) are approximated by a special finite difference scheme to maintain the accuracy. Then an iterative algorithm is introduced, where transmission of the information to the neighbours is done using the piecewise-linear interpolation. We discuss parameter-uniform convergence of the constructed method using auxiliary problems that separates the discretization and iteration errors. Note that the convergence analysis of domain decomposition methods for problems with Dirichlet boundary conditions cannot be straightforwardly applied to the present method due to the presence of the Robin boundary conditions. Firstly, we require a different definition of auxiliary problems to handle the Robin boundary conditions. The boundary and initial conditions of the auxiliary problems in earlier works are simply defined using the exact solution of the problem, but that is not entirely true for the present problem. Secondly, we require a more complex constant coefficient problem (in Theorem 3.2) and altogether different barrier functions (in Theorems 3.2 and 3.3) for bounding the error between the auxiliary solution and the Schwarz iterates. It is proved that the resulting numerical approximations are parameter-uniform and, more interestingly, that the iteration convergence is optimal for small \(\varepsilon .\) Finally, numerical results are provided to support the convergence theory.

The rest of the article is organized in different sections as follows. Section 2 provides a priori bounds for the solution derivatives, and Sect. 3 includes the development of the SWR method for problem (1)–(3). Section 4 includes the convergence analysis of the developed method. Finally, in Sect. 5, we present the numerical results for two test examples confirming our convergence theory.

Notation: C denotes a generic positive constant which is independent of the parameters N,  M, \( \varepsilon ,\) and k. For \( v \in C({\overline{\Omega }}),\) let us define \( v_{i,j}=v(x_i,t_j)\). Here, \( \left\Vert . \right\Vert _Q \) is used to define the maximum norm on a closed and bounded set Q,  and \( \left\Vert . \right\Vert _{Q^{N,M}} \) is used to define the corresponding discrete maximum norm.

2 Asymptotic behavior

We discuss a priori bounds on the solution derivatives of (1)–(3). The existence of a unique solution of problem (1)–(3) is established in the following lemma.

Lemma 1

Suppose \(a, f\in C^{(\beta ,\beta /2)}(\overline{\Omega }),\) \(g_\ell ,~g_r\in C^{\frac{1+\beta }{2}}([0,T])\), \(g_b\in C^{2+\beta }(\overline{{\textsf{D}}})\), \(\beta \in (0,1),\) and the compatibility conditions up to the first order are true. Then, problem (1)–(3) has a unique solution \(u\in C^{(2+\beta ,1+\beta /2)}(\overline{\Omega }).\)

Proof

See [23, Chapter IV, Section 5]). \(\square \)

In the following lemma, crude bounds for the derivatives of u are given.

Lemma 2

Suppose \(a, f\in C^{(2+\beta ,1+\beta /2)}(\overline{\Omega }),\) \(g_\ell ,~g_r\in C^{\frac{3+\beta }{2}}([0,T])\), \(g_b\in C^{4+\beta }(\overline{{\textsf{D}}})\), \(\beta \in (0,1),\) and the compatibility conditions up to the second order are true. Then, problem (1)–(3) has a unique solution \(u\in C^{(4+\beta ,2+\beta /2)}(\overline{\Omega }).\) Moreover, it holds

$$\begin{aligned} \left\Vert \frac{\partial ^{q_1+q_2}u}{\partial x^{q_1}\partial t^{q_2}}\right\Vert _{{\overline{\Omega }}}\le C\varepsilon ^{-q_1 /2}~~~\text{ for }~~0\le q_1+2q_2 \le 4, \end{aligned}$$
(4)

Proof

The proof follows from the arguments in [37, Theorem 5]. \(\square \)

The above bounds are not enough for the convergence analysis in Sect. 4. We require to split u into regular and singular parts, and bounds for their derivatives as given in the following lemma.

Lemma 3

Suppose \(a, f\in C^{(4+\beta ,2+\beta /2)}(\overline{\Omega }),\) \(g_\ell ,~g_r\in C^{\frac{5+\beta }{2}}([0,T])\), \(g_b\in C^{6+\beta }(\overline{{\textsf{D}}})\), \(\beta \in (0,1),\) and sufficient compatibility conditions hold. Then, one can split u as \(u=u_1+u_2,\) where \(u_1\) and \(u_2\) are the regular and singular parts, respectively, satisfying

$$\begin{aligned}{} & {} \displaystyle |\partial _x^{q_1} u_1(x,t)|\le C(1+\varepsilon ^{(2-q_1)/2}),~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(5)
$$\begin{aligned}{} & {} ~~~~|\partial _x^{q_1} u_2(x,t)|\le C\varepsilon ^{-{q_1}/2}\left( e^{(-x\sqrt{\alpha /\varepsilon })}+e^{(-(1-x)\sqrt{\alpha /\varepsilon })}\right) , \end{aligned}$$
(6)

for \((x,t)\in {\overline{\Omega }},\) \(0\le q_1\le 4.\)

Proof

The lemma can be proved following the arguments in [37, Theorem 6]. \(\square \)

3 The domain decomposition method

We partition \(\Omega \) into \(\Omega _p={\textsf{D}}_p\times (0,T],~p=\ell ,m,r,\) where \({\textsf{D}}_\ell =(0,2\rho ),~{\textsf{D}}_m=(\rho ,1-\rho ),~{\textsf{D}}_r=(1-2\rho ,1)\) with the parameter \(\rho \) defined by

$$\begin{aligned} \rho = \text{ min }\left\{ \frac{1}{4},~~2\sqrt{\frac{\varepsilon }{\alpha }}\, \ln N\right\} . \end{aligned}$$
(7)

Here, \(\rho \) is chosen as the Shishkin transition parameter [9, 26], which enables us to have fine meshes (for small \(\varepsilon \)) on \(\Omega _\ell \) and \(\Omega _r\) (the subdomains corresponding to the boundary layers). Further, we discretize each subdomain \(\Omega _p\) with a uniform mesh in both the spatial and temporal directions. On each spatial subdomain \(\overline{{\textsf{D}}}_p=[a,b],\) we define a uniform mesh \(\overline{{\textsf{D}}}_p^{N}=\{x_{p;i}=a+ih_p, i=0,\dots ,N,~h_p=(b-a)/N\}.\) When the domain is obvious, for convenience, the term p will be dropped from the subscript in \(x_{p;i}.\) On the interval \({\overline{\Omega }}_t=[0,T],\) we define a uniform mesh \({\overline{\Omega }}_t^{M}=\{t_j=j\Delta t,~j=0,\dots ,M,~ \text{ with }~\Delta t=T/M\}.\) Here, M and N are discretization parameters in time and space directions respectively. Defining \({\textsf{D}}^N_p=\overline{{\textsf{D}}}_p^N\cap {\textsf{D}}_p,\) and \(\Omega _t^M={\overline{\Omega }}_t^M\cap (0,T],\) the mesh \(\Omega _p^{N,M}\) corresponding to \(\Omega _p\) is defined as \(\Omega _p^{N,M}={\textsf{D}}_p^N\times \Omega _t^M.\) Further, we define \({\bar{\Omega }}_p^{N,M}=\bar{{\textsf{D}}}_p^N\times {\bar{\Omega }}_t^M\) and \(\gamma _{p,\ell }^{N,M}=\{x_{p;0}\}\times \Omega _t^M,\) \(\gamma _{p,r}^{N,M}=\{x_{p;N}\}\times \Omega _t^M,\) \(\gamma _{p,b}^{N,M}=\overline{{\textsf{D}}}_p^{N}\times \{t_0\}.\) Then, on each \(\Omega _p^{N,M},\, p=\ell ,m,r\), the discretization of (1) is defined as follows

$$\begin{aligned} L^{N,M}_pU_{p;i,j}:= {B_t^-}U_{p;i,j}-\varepsilon \delta _x^2 U_{p;i,j}+a_{i,j}U_{p;i,j}= f_{i,j},~~(x_{p;,i},t_j)\in \Omega _p^{N,M}, \end{aligned}$$

where

$$\begin{aligned}{} & {} \delta _x^2U_{p;i,j}=\frac{[F_x^+U_p-F_x^-U_p]_{i,j}}{h_p},\\{} & {} F_x^+U_{p;i,j}=(U_{p;i+1,j}-U_{p;i,j})/h_p, \quad F_x^-U_{p;i,j}=(U_{p;i,j}-U_{p;i-1,j})/h_p,~~\\{} & {} \text{ and }~~{B_t^-}U_{p;i,j}=(U_{p;i,j}-U_{p;i,j-1})/{\Delta t}. \end{aligned}$$

We discretize (2) as follows

$$\begin{aligned} \left\{ \begin{array}{ll} ~\Gamma _\ell ^{N,M}U_{\ell }(0,t_j):= U_{\ell }(0,t_j)-\sqrt{\varepsilon }F_x^+U_{\ell }(0,t_j)+\frac{h_{\ell }}{2 \sqrt{\varepsilon }}(aU_{\ell }+{B_t^-}U_{\ell })(0,t_j)\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= g_\ell (t_j)+\frac{h_{\ell }}{2\sqrt{\varepsilon }}f(0,t_j),~~~~t_j\in \Omega _t^M,\\[8pt] ~\Gamma _r^{N,M}U_r(1,t_j):= U_r(1,t_j)+\sqrt{\varepsilon }F_x^-U_r(1,t_j)+\frac{h_r}{2\sqrt{\varepsilon }}(aU_r+{B_t^-}U_r)(1,t_j)\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\,=g_r(t_j)+\frac{h_r}{2\sqrt{\varepsilon }}f(1,t_j),~~~~t_j\in \Omega _t^M.\\ \end{array} \right. \end{aligned}$$
(8)

Note that we have approximated the boundary conditions using a special discretization scheme. If we had used the standard upwind scheme for the discretization of the boundary conditions we would have obtained only first order accuracy. Therefore, we use a special discretization scheme for the boundary conditions, which is based on improving the truncation error to maintain second order accuracy.

Consider \({\overline{\Omega }}^{N,M}=({\overline{\Omega }}^{N,M}_{\ell }{\setminus } {\overline{\Omega }}_{m})\bigcup {\overline{\Omega }}_m^{N,M}\bigcup ({\overline{\Omega }}^{N,M}_{r}{\setminus } {\overline{\Omega }}_{m}).\) To compute as the approximate solution of problem (1)–(3) the algorithm is defined as follows. We start the iterative process with \(U^{[0]}(x_{i},t_j)\) as the initial approximation, defined as follows: \(U^{[0]}(x_{i},0)=g_b(x_i), 0\le x_{i} \le 1;\) \(U^{[0]}(x_{i},t_j)=0,\) \(0< x_{i}< 1,\,0<t_j\le T.\) Suppose \({\mathcal {I}}_j Z\) is used to denote the piecewise-linear interpolant function of Z at \(t_j.\) For each \(k\ge 1,\) we solve

$$\begin{aligned} \left\{ \begin{array}{ll} {L}^{N,M}_\ell U_\ell ^{[k]}=f &{} \text{ in }~ \Omega _\ell ^{N,M}, \\ \Gamma _\ell ^{N,M}U_\ell ^{[k]}(0,t_j)=g_\ell (t_j)+\frac{h_\ell }{2\sqrt{\varepsilon }}f(0,t_j) &{} \text{ if }~t_j\in \Omega _t^M,\\ U_\ell ^{[k]}(2\rho ,t_j)={\mathcal {I}}_jU^{[k-1]}(2\rho ,t_j) &{} \text{ if }~t_j\in \Omega _t^M,~~~~~~~~~~~~~~~~~~~~~\, \\ U_\ell ^{[k]}(x_i,0)=g_b(x_i) &{} \text{ if }~ x_i\in \overline{{\textsf{D}}}_\ell ^{N} ~,\\ \end{array} \right. \\ ~~~\left\{ \begin{array}{ll} {L}^{N,M}_r U_r^{[k]}=f &{} \text{ in }~\Omega _r^{N,M}, \\ U_r^{[k]}(1-2\rho ,t_j)={\mathcal {I}}_jU^{[k-1]}(1-2\rho ,t_j) &{} \text{ if }~t_j\in \Omega _t^M,~~~~~~~~~~~~~~~~~~~~ \\ \Gamma _r^{N,M}U_r^{[k]}(1,t_j)=g_r(t_j)+\frac{h_r}{2\sqrt{\varepsilon }}f(1,t_j) &{} \text{ if }~t_j\in \Omega _t^M,\\ U_r^{[k]}(x_i,0)=g_b(x_i) &{} \text{ if }~ x_i\in \overline{{\textsf{D}}}_r^{N} ~,\\ \end{array} \right. ~~~~ \\ \left\{ \begin{array}{ll} {L}^{N,M}_mU_m^{[k]}=f &{} \text{ in }~ \Omega _m^{N,M}, \\ U_m^{[k]}(\rho ,t_j)={\mathcal {I}}_jU^{[k]}_\ell (\rho ,t_j),&{}\text{ for }~t_j\in \Omega _t^M,\\ U_m^{[k]}(1-\rho ,t_j)={\mathcal {I}}_jU^{[k]}_r(1-\rho ,t_j) &{} \text{ for }~t_j\in \Omega _t^M, \\ U_m^{[k]}(x_i,0)=g_b(x_i) &{} \text{ for }~ x_i\in \overline{{\textsf{D}}}_m^{N}.\\ \end{array} \right. ~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$

We then compute \(U^{[k]}\) by

$$\begin{aligned} ~~~~~~ U^{[k]} = \left\{ \begin{array}{ll} U^{[k]}_\ell ~~~\text{ in }~{\overline{\Omega }}_\ell ^{N,M}\setminus {\overline{\Omega }}_m,\\[8pt] U^{[k]}_m ~~~\text{ in }~{\overline{\Omega }}_m^{N,M}, \\[8pt] U^{[k]}_r ~~~\text{ in }~{\overline{\Omega }}_r^{N,M}\setminus {\overline{\Omega }}_m. \end{array} \right. \end{aligned}$$
(9)

The iterations are performed until

$$\begin{aligned}||U^{[k+1]}-U^{[k]}||_{{\overline{\Omega }}^{N,M}}\le tol\end{aligned}$$

is reached, where tol is a specified user tolerance.

For \(p=\ell ,~r,~m,\) suppose the operator \({\mathcal {T}}_p^{N,M}\) is defined as \({\mathcal {T}}_p^{N,M}Z(y,t_j)=Z(y,t_j), ~y=\rho ,~2\rho ,~1-\rho ,~1-2\rho ,t_j\in \Omega _t^M\) and \({\mathcal {T}}_\ell ^{N,M}Z(0,t_j)=\Gamma _\ell ^{N,M}Z(0,t_j),~{\mathcal {T}}_r^{N,M}Z(1,t_j)=\Gamma _r^{N,M}Z(1,t_j),~t_j\in \Omega _t^M.\) Now using the arguments in [37, Theorem 7] we can prove the following maximum principle for \(L_p^{N,M}.\)

Lemma 4

Suppose \(Z_p,~p=m,r,\ell ,\) satisfies \(Z_p(x_i,0)\ge 0\) for \(x_i\in \overline{{\textsf{D}}}^{N}_p\) and \({\mathcal {T}}_{p}^{N,M}Z_p(x_0,t_j)\ge 0,\) \(~{\mathcal {T}}_p^{N,M}Z_p(x_N,t_j)\ge 0,\) \(t_j\in \Omega _t^M.\)

Then, if \(L_p^{N,M}Z_{p;i,j}\ge 0\) in \(\Omega _p^{N,M},\) it holds that \(Z_{p;i,j}\ge 0\) in \( {\overline{\Omega }}_p^{N,M}.\)

The stability of the numerical scheme is given by the following lemma.

Lemma 5

For any mesh function \(Z_p,\) we have

$$\begin{aligned}||Z_p||_{\overline{\Omega }_p^{N,M}}\le \max \left\{ ||{\mathcal {T}}_{p}^{N,M}Z_p||_{\gamma _{p,\ell }^{N,M}},||{\mathcal {T}}_{p}^{N,M}Z_p||_{\gamma _{p,r}^{N,M}},||Z_p||_{\gamma _{p,b}^{N,M}},\dfrac{1}{\alpha }||L_p^{N,M}Z_p||_{\Omega _p^{N,M}}\right\} .\end{aligned}$$

Proof

Suppose \(\Phi ^{\pm }(x_i,t_j)={\Theta }\pm Z(x_i,t_j)\) is a mesh function with

$$\begin{aligned}{\Theta }=\max \left\{ ||{\mathcal {T}}_{p}^{N,M}Z_p||_{\gamma _{p,\ell }^{N,M}},||{\mathcal {T}}_{p}^{N,M}Z_p||_{\gamma _{p,r}^{N,M}},||Z_p||_{\gamma _{p,b}^{N,M}},\dfrac{1}{\alpha }||L_p^{N,M}Z_p||_{\Omega _p^{N,M}}\right\} .\end{aligned}$$

Then, it is easy to verify that \({\mathcal {T}}_\ell ^{N,M}\Phi ^{\pm }(x_0,t_j)\ge 0\), \({\mathcal {T}}_r^{N,M}\Phi ^{\pm }(x_N,t_j)\ge 0,\) \(t_j\in \Omega _t^M,\) \(\Phi ^{\pm }(x_i,0)\ge 0\) for \(x_i\in \overline{{\textsf{D}}}^{N}_p,\) and \(L_p^{N,M}\Phi ^{\pm }(x_i,t_j)\ge 0,\) \((x_i,t_j)\in \Omega _p^{N,M}\). Hence, from Lemma 4, the proof follows. \(\square \)

4 Convergence analysis

We now establish that the method gives parameter-uniform approximations to the solution of problem (1)–(3). Further, we prove that the iterative process converges optimally for small \(\varepsilon .\) We consider the following auxiliary mesh function

$$\begin{aligned} ~~~~~~ {\overline{U}} = \left\{ \begin{array}{ll} {\overline{U}}_\ell ~~~\text{ in }~{\overline{\Omega }}_\ell ^{N,M}\setminus {\overline{\Omega }}_m,\\[8pt] {\overline{U}}_m ~~~\text{ in }~{\overline{\Omega }}_m^{N,M}, \\[8pt] {\overline{U}}_r ~~~\text{ in }~{\overline{\Omega }}_r^{N,M}\setminus {\overline{\Omega }}_m, \end{array} \right. \end{aligned}$$
(10)

where \({\overline{U}}_p,\) \(p=\ell ,m,r,\) are such that

$$\begin{aligned} \left\{ \begin{array}{ll} {L}^{N,M}_\ell {\overline{U}}_\ell =f &{} \text{ in }~\Omega _\ell ^{N,M}, \\ \Gamma _\ell ^{N,M}{\overline{U}}_\ell (0,t_j)=\Gamma _\ell ^{N,M}U^{[k]}(0,t_j)&{} \text{ for }~t_j\in \Omega _t^M,\\ {\overline{U}}_\ell (2\rho ,t_j)=u(2\rho ,t_j) &{} \text{ for }~t_j\in \Omega _t^M,~~~~~~~~~ \\ {\overline{U}}_\ell (x_i,0)=u(x_i,0) &{} \text{ if }~ x_i\in \overline{{\textsf{D}}}_{\ell }^{N},\\ \end{array} \right. \\ \left\{ \begin{array}{ll} {L}^{N,M}_m {\overline{U}}_m=f &{} \text{ in }~\Omega _m^{N,M}, \\ {\overline{U}}_m(\rho ,t_j)=u(\rho ,t_j)&{} \text{ for }~t_j\in \Omega _t^M, \\ {\overline{U}}_m(1-\rho ,t_j)=u(1-\rho ,t_j) &{} \text{ for }~t_j\in \Omega _t^M, \\ {\overline{U}}_m(x_i,0)=u(x_i,0) &{} \text{ if }~ x_i\in \overline{{\textsf{D}}}_{m}^{N},\\ \end{array} \right. ~~~~~~~~~~~~~ \\ \left\{ \begin{array}{ll} {L}^{N,M}_r {\overline{U}}_r=f &{} \text{ in }~\Omega _r^{N,M}, \\ {\overline{U}}_r(1-2\rho ,t_j)=u(1-2\rho ,t_j) &{} \text{ for }~t_j\in \Omega _t^M, \\ \Gamma _r^{N,M}{\overline{U}}_r(1,t_j)=\Gamma _r^{N,M}U^{[k]}(1,t_j) &{} \text{ for }~t_j\in \Omega _t^M, \\ {\overline{U}}_r(x_i,0)=u(x_i,0) &{} \text{ if }~ x_i\in \overline{{\textsf{D}}}_{r}^{N}.\\ \end{array} \right. ~~~~~~~~~~ \end{aligned}$$

Here, the discrete operators \({L}^{N,M}_\ell , {L}^{N,M}_m, {L}^{N,M}_r, \Gamma _r^{N,M},\) and \(\Gamma _{\ell }^{N,M}\) are the ones that were defined in the previous section. The main difference between these problems and the problems in the previous section is that we used the solution u of problem (1)–(3) in the boundary conditions for \({\overline{U}}_{\ell }\) at \((2\rho ,t_j),\) \({\overline{U}}_{r}\) at \((1-2\rho ,t_j),\) and \({\overline{U}}_{m}\) at \((\rho ,t_j)\) and \((1-\rho ,t_j).\) By the triangle inequality, we get

$$\begin{aligned} ||u-U^{[k]}||_{{\overline{\Omega }}^{N,M}}\le ||u-{\overline{U}}||_{{\overline{\Omega }}^{N,M}} +||{\overline{U}}-U^{[k]}||_{{\overline{\Omega }}^{N,M}}. \end{aligned}$$
(11)

The following lemma gives the bound of the first term on the LHS of the inequality in (11).

Lemma 6

Suppose u is the solution of (1)–(3) and \({\overline{U}}\) is the auxiliary mesh function defined in (10). Then

$$\begin{aligned} ||u-{\overline{U}}||_{{\overline{\Omega }}^{N,M}}\le C(\Delta t + (N^{-1}\ln N)^2). \end{aligned}$$
(12)

Proof

Note that \((u-\overline{U_\ell })(2\rho ,t_j)=0\) for \( t_j \in \Omega _t^M.\) On the other hand, from (1)–(2) and (8) it follows that

$$\begin{aligned}{} & {} \Gamma _{\ell }^{N,M}(u-{\overline{U}}_{\ell })(0,t_j) =\left( u-\sqrt{\varepsilon }F_x^+u+\frac{h_\ell }{2\sqrt{\varepsilon }}(au+{B_t^-}u)-(u-\sqrt{\varepsilon }u_x)-\frac{h_\ell }{2\sqrt{\varepsilon }}f\right) (0,t_j)\\{} & {} =\left( \sqrt{\varepsilon }(u_x-F_x^+u)+\frac{h_\ell }{2 \sqrt{\varepsilon }}(au-f+u_t)\right) (0,t_j)+\frac{h_\ell }{2 \sqrt{\varepsilon }}({B_t^-}u-u_t)(0,t_j)\\{} & {} =\sqrt{\varepsilon }\left( u_x-F_x^+u+\frac{h_\ell }{2}u_{xx}\right) (0,t_j)+\frac{h_\ell }{2 \sqrt{\varepsilon }}({B_t^-}u-u_t)(0,t_j),~~t_j \in \Omega _t^{M}. \end{aligned}$$

Using Taylor expansions and (4) we get

$$\begin{aligned} \left| \Gamma _\ell ^{N,M}(u-\overline{U_\ell })(0,t_j)\right|\le & {} \frac{\sqrt{\varepsilon }}{6} {h}_\ell ^2 \left\Vert u_{xxx}(.,t_j)\right\Vert _{[x_{0},x_{1}]}+\frac{h_\ell }{4\sqrt{\varepsilon }}(t_j-t_{j-1})\left\Vert u_{tt}(x_i,.)\right\Vert _{[t_{j-1},t_j]}\\\le & {} C(\Delta t+(N^{-1}\ln N)^2). \end{aligned}$$

Now

$$\begin{aligned} L^{N,M}_p(u-{\overline{U}}_p)=(L^{N,M}_p-L)u=\left( {B_t^-}u-u_t\right) -\varepsilon \left( \delta _x^2u-u_{xx}\right) ~~\text{ in }~\Omega ^{N,M}_p,~p=\ell ,m,r, \end{aligned}$$
(13)

and

$$\begin{aligned} \left| L^{N,M}_p(u-{\overline{U}}_p)_{i,j}\right| \le \frac{(t_j-t_{j-1})}{2}\left\Vert {u_{tt}(x_i,.)} \right\Vert _{[t_{j-1},t_j]}+\frac{\varepsilon }{12}h^2_p\left\Vert {u_{xxxx}(.,t_j)}\right\Vert _{[x_{i-1},x_{i+1}]}, \end{aligned}$$
(14)

where we have used Taylor expansions to get (14). Since \(h_\ell \le C\sqrt{\varepsilon }N^{-1}\ln N,\) using (4) it follows from (14) that

$$\begin{aligned} \left| L^{N,M}_\ell (u-{\overline{U}}_\ell )_{i,j}\right| \le C(\Delta t +(N^{-1}\ln N)^2) ~~~\text{ in }~\Omega ^{N,M}_\ell . \end{aligned}$$

Hence, by Lemma 4 with \(C(\Delta t+(N^{-1}\ln N)^2) \pm (u-{\overline{U}}_\ell )_{i,j},\) one gets

$$\begin{aligned} ||u-{\overline{U}}_{\ell }||_{{\overline{\Omega }}^{N,M}_{\ell }}\le C(\Delta t + (N^{-1}\ln N)^2).\end{aligned}$$

A similar analysis gives

$$\begin{aligned} ||u-{\overline{U}}_{r}||_{{\overline{\Omega }}^{N,M}_{r}}\le C(\Delta t + (N^{-1}\ln N)^2).\end{aligned}$$

Suppose \( \rho =1/4.\) Then, we have \(h_m=0.5N^{-1}\) and \(\varepsilon ^{-1}\le C\ln ^2 N.\) Again, using (4) it follows from (14) that

$$\begin{aligned}\left| L^{N,M}_m(u-{\overline{U}}_m)_{i,j}\right| \le C(\Delta t + (N^{-1}\ln N)^2)~~~\text{ in }~\Omega ^{N,M}_m.\end{aligned}$$

If \(\rho =2\sqrt{\frac{\varepsilon }{\alpha }}\ln N,\) then \(h_m\le CN^{-1}.\) For \(({B_t^-}u-u_t)\) use Taylor expansion and (4) to get

$$\begin{aligned}\left| \left( {B_t^-}u-u_t\right) _{i,j}\right| \le C\Delta t ~~~\text{ in }~ \Omega ^{N,M}_m.\end{aligned}$$

For \(\varepsilon (\delta _x^2u-u_{xx})\) use the decomposition \(u=u_1+u_2,\) Taylor expansions, and (5)-(6) to get

$$\begin{aligned} \varepsilon \left| \left( \delta _x^2u-u_{xx}\right) _{i,j}\right|\le & {} \varepsilon \left( \left| \left( \delta _x^2u_1-\frac{\partial ^2 u_{1}}{\partial x^2}\right) _{i,j}\right| +\left| \left( \delta _x^2u_2-\frac{\partial ^2 u_{2}}{\partial x^2}\right) _{i,j}\right| \right) \\\le & {} C\varepsilon h^2_m\left\Vert \frac{\partial ^4 u_{1}(.,t_j)}{\partial x^4} \right\Vert _{[x_{i-1},x_{i+1}]}+C\varepsilon \left\Vert \frac{\partial ^2 u_{2}(.,t_j)}{\partial x^2}\right\Vert _{[x_{i-1},x_{i+1}]}\\\le & {} CN^{-2}+2Ce^{-\rho \sqrt{\alpha /\varepsilon }} \le CN^{-2}. \end{aligned}$$

Thus, \(\left| L^{N,M}_m(u-{\overline{U}}_m)_{i,j}\right| \le C(\Delta t + (N^{-1}\ln N)^2)~~ \text{ in } ~\Omega ^{N,M}_m.\) So, applying Lemma 4 to \(C(\Delta t+(N^{-1}\ln N)^2) \pm (u-{\overline{U}}_m),\) one gets

$$\begin{aligned} ||u-{\overline{U}}_{m}||_{{\overline{\Omega }}^{N,M}_m}\le C(\Delta t + (N^{-1}\ln N)^2).\end{aligned}$$

This completes the proof. \(\square \)

The following notation will be used in the next two theorems.

$$\begin{aligned} \mu ^{[k]}= & {} \max \left\{ \max _{t_j\in \Omega _t^M}|({\overline{U}}_{\ell }-{\mathcal {I}}_jU^{[k-1]})(2\rho ,t_j)|, \max _{t_j\in \Omega _t^M}|({\overline{U}}_{r}-{\mathcal {I}}_jU^{[k-1]})(1-2\rho ,t_j)|\right\} ,\\ \mu _{\rho }= & {} \displaystyle \max \left\{ \max _{t_j\in \Omega _t^M}|({\overline{U}}_{m}-{\overline{U}}_{\ell })(\rho ,t_j)|, \max _{t_j\in \Omega _t^M}|({\overline{U}}_{m}-{\overline{U}}_{r})(1-\rho ,t_j)| \right\} . \end{aligned}$$

We will show the parameter-uniform convergence of the method in two cases: when \(\rho =2\sqrt{\frac{\varepsilon }{\alpha }}\ln N\) and \(\rho =1/4\). In the next theorem, for \(\rho =2\sqrt{\frac{\varepsilon }{\alpha }}\ln N,\) we first obtain a bound of \(||{\overline{U}}-U^{[1]}||_{{\overline{\Omega }}^{N,M}}\) and then combine it with Lemma 6 to get a bound of \(||u-U^{[1]}||_{{\overline{\Omega }}^{N,M}}.\) So, this result demonstrates that only one iteration is enough to achieve the required accuracy.

Theorem 1

Suppose u is the exact solution of (1)–(3) and \(U^{[1]}\) is its approximation obtained after the first iterate of the proposed method. Then, for \(\rho =2\sqrt{\frac{\varepsilon }{\alpha }}\ln N,\) it holds

$$\begin{aligned}||u-U^{[1]}||_{{\overline{\Omega }}^{N,{M}}}\le C(\Delta t +(N^{-1}\ln N)^2).\end{aligned}$$

Proof

Observe that

$$\begin{aligned} \left\{ \begin{array}{ll} L^{N,M}_\ell ( {\overline{U}}_\ell -U_\ell ^{[1]})=0 &{} \text{ in }~ \Omega _\ell ^{N,M}, \\ ({\overline{U}}_\ell -U_\ell ^{[1]})(x_i,0)=0 &{} \text{ for }~ x_i\in \overline{{\textsf{D}}}^{N}_{\ell },\\ \Gamma _\ell ^{N,M}({\overline{U}}_\ell -U_\ell ^{[1]})(0,t_j)=0 &{} \text{ for }~t_j\in \Omega _t^{M},\\ |({\overline{U}}_\ell -U_\ell ^{[1]})(2\rho ,t_j)|\le \mu ^{[1]} &{} \text{ for }~t_j\in \Omega _t^{M}.~~~~~~~~~~~~ \\ \end{array} \right. \end{aligned}$$

For an arbitrary mesh function Y,  let us define

$$\begin{aligned}{\widetilde{\Gamma }}_\ell ^{N,M} Y(0,t_j):=Y(0,t_j)-\sqrt{\varepsilon }F_x^+Y(0,t_j)+\frac{h_{\ell }}{2 \sqrt{\varepsilon }}(\alpha Y+{B_t^-}Y)(0,t_j).\end{aligned}$$

For \((x_i,t_j)\in {\overline{\Omega }}_\ell ^{N,{M}},\) consider

$$\begin{aligned}\chi ^\pm (x_i,t_j)=\left[ \chi _\ell \pm ({\overline{U}}_\ell -U_\ell ^{[1]})\right] (x_i,t_j),\end{aligned}$$

where \(\chi _\ell \) solves

$$\begin{aligned} \left\{ \begin{array}{ll} {B_t^-}\chi _\ell -\varepsilon \delta ^2_x \chi _\ell +\alpha \chi _\ell =0 &{} \text{ in }~\Omega _\ell ^{N,{M}}, \\ \chi _\ell (x_i,0)=\mu ^{[1]} \frac{B\xi _1^i-A\xi _2^i}{B\xi _1^N-A\xi _2^N} &{} \text{ for }~ x_i\in \overline{{\textsf{D}}}_{\ell }^{N},\\ {\widetilde{\Gamma }}_\ell ^{N,M} \chi _\ell (0,t_j)=0 &{} \text{ for }~t_j\in \Omega _t^{M},\\ \chi _\ell (2\rho ,t_j)= \mu ^{[1]} &{} \text{ for }~t_j\in \Omega _t^{M}\\ \end{array} \right. \end{aligned}$$
(15)

with

$$\begin{aligned}A=2\sqrt{\varepsilon }h_\ell -2\varepsilon (\xi _1-1)+h_\ell ^2a ~,~ B=2\sqrt{\varepsilon }h_\ell -2\varepsilon (\xi _2-1)+h_\ell ^2a,\end{aligned}$$

and \(\xi _i, i=1,2,\) are as follows

$$\begin{aligned}\xi _1=\lambda _1+\lambda _2~~~\text{ and }~~~\xi _2=\lambda _1-\lambda _2\end{aligned}$$

with

$$\begin{aligned}\lambda _1=1+\left( \frac{\rho }{N}\sqrt{\frac{\alpha }{\varepsilon }}~\right) ^2,~\lambda _2=2\left( \frac{\rho }{N}\sqrt{\frac{\alpha }{\varepsilon }}~\right) \sqrt{1+\left( \frac{\rho }{N}\sqrt{\frac{\alpha }{\varepsilon }}~\right) ^2}.\end{aligned}$$

The solution of (15) is given by

$$\begin{aligned} {\chi }_{\ell }(x_i,t_j)=\mu ^{[1]}\frac{\varphi \xi _1^i- \xi _2^{i}}{\varphi \xi _1^N-\xi _2^N}, ~~\varphi =B/A, \end{aligned}$$
(16)

which is monotonically increasing and satisfies \({\chi }_{\ell }\ge 0~ \text{ in }~ {\overline{\Omega }}_\ell ^{N,M}.\) Thus, one gets

$$\begin{aligned}\chi ^\pm (x_i,0)\ge 0~~\text{ if }~~x_i\in \overline{{\textsf{D}}}^N_\ell ,~~~~\Gamma _\ell ^{N,M}\chi ^\pm (0,t_j)\ge 0, ~\chi ^\pm (2\rho ,t_j)\ge 0~~\text{ if }~t_j\in \Omega _t^{M},\end{aligned}$$

and \(L^{N,M}_\ell \chi ^\pm (x_i,t_j)\ge 0\) if \((x_i,t_j)\in \Omega _\ell ^{N,M}.\) Hence, applying Lemma 4 to \(\chi ^\pm ,\) we obtain

$$\begin{aligned}|({\overline{U}}_\ell -U_\ell ^{[1]})_{i,j}|\le \chi _\ell (x_i,t_j), ~~~\text{ if }~~(x_i,t_j)\in {\overline{\Omega }}_\ell ^{N,M}.\end{aligned}$$

Further, since \(x_i\le \rho \) for \((x_i,t_j)\in {\overline{\Omega }}_\ell ^{N,M}\backslash {\overline{\Omega }}_m,\) it follows from (16) that

$$\begin{aligned}{\chi }_{\ell }(x_i,t_j)\le \mu ^{[1]}\frac{\varphi \xi _1^{N/2}- \xi _2^{N/2}}{\varphi \xi _1^N-\xi _2^N}=\mu ^{[1]}\frac{\varphi ^2\xi _1^N-\xi _2^{N}}{(\varphi \xi _1^N-\xi _2^N)(\varphi \xi _1^{N/2}+\xi _2^{N/2})}.\end{aligned}$$

This implies that

$$\begin{aligned}{\chi }_{\ell }(x_i,t_j)\le \frac{\mu ^{[1]}}{\varphi \xi _1^{N/2}+\xi _2^{N/2}}\left[ \frac{\varphi ^2-\xi _2^{N}/\xi _1^N}{\varphi -\xi _2^{N}/\xi _1^N}\right] .\end{aligned}$$

Now, since \(\varphi >1\) and \(\xi _2/\xi _1<1,\) we have \((\varphi ^2-\xi _2^{N}/\xi _1^N)/(\varphi -\xi _2^{N}/\xi _1^N)\le C\varphi .\) Thus, one gets

$$\begin{aligned}{\chi }_{\ell }(x_i,t_j)\le \mu ^{[1]}\frac{C\varphi }{\varphi \xi _1^{N/2}+\xi _2^{N/2}}\le \frac{C\mu ^{[1]}}{\xi _1^{N/2}}.\end{aligned}$$

For \(\rho =2{\varepsilon }^{1/2}{\alpha }^{-1/2}\ln N,\) it holds

$$\begin{aligned}\xi _1^{-N/2}\le \left( 1+\frac{\rho }{N}\sqrt{\frac{\alpha }{\varepsilon }}~\right) ^{-N}\le 4N^{-2},~~\text{ if }~N\ge 1,\end{aligned}$$

having used \(\lambda _2\ge 2\left( \frac{\rho }{N}\sqrt{\frac{\alpha }{\varepsilon }}~\right) \) and the arguments in [26, Lemma 5.1]. Further, since \(\mu ^{[1]}\le C,\) we have \({\chi }_{\ell }(x_i,t_j)\le C N^{-2}\,\) in \({\overline{\Omega }}_\ell ^{N,M}\backslash {\overline{\Omega }}_m.\) Thus, we have

$$\begin{aligned} ||{\overline{U}}_\ell -U^{[1]}_\ell ||_{{\overline{\Omega }}^{N,M}_\ell \backslash {\overline{\Omega }}_m}\le CN^{-2}. \end{aligned}$$
(17)

Similarly, it is

$$\begin{aligned} ||{\overline{U}}_r-U^{[1]}_r||_{{\overline{\Omega }}^{N,M}_r\backslash {\overline{\Omega }}_m}\le CN^{-2}. \end{aligned}$$
(18)

Next, consider \({\overline{U}}_m-U_m^{[1]}\) which satisfies

$$\begin{aligned}{} & {} L^{N,M}_m({\overline{U}}_m-U_m^{[1]})=0~~~ \text{ in }~\Omega _m^{N,M},\\{} & {} ({\overline{U}}_m-U_m^{[1]})(x_i,0)=0,~~~ \text{ if }~ x_i\in \overline{{\textsf{D}}}_{m}^{N},~~~~~~\\{} & {} |({\overline{U}}_m-U_m^{[1]})(\rho ,t_j)|=|({\overline{U}}_m-{\mathcal {I}}_jU_\ell ^{[1]})(\rho ,t_j)|~~~~~~~~~~~~~~~~~~~~~~~~~\\{} & {} \le |({\overline{U}}_m-{\overline{U}}_\ell )(\rho ,t_j)|+|({\overline{U}}_{\ell }-U_\ell ^{[1]})(\rho ,t_j)|\\{} & {} \le \mu _{\rho }+CN^{-2},~~~t_j\in \Omega _t^M,\end{aligned}$$

and

$$\begin{aligned}{} & {} |({\overline{U}}_m-U_m^{[1]})(1-\rho ,t_j)|=|({\overline{U}}_m-{\mathcal {I}}_jU_r^{[1]})(1-\rho ,t_j)| \le |({\overline{U}}_m-{\overline{U}}_r)(1-\rho ,t_j)|\\{} & {} \quad +|({\overline{U}}_r-U_r^{[1]})(1-\rho ,t_j)|\\{} & {} \le \mu _{\rho }+CN^{-2}~~\text{ for }~t_j\in \Omega _t^M,\end{aligned}$$

since the mesh points \((\rho ,t_j)\) and \((1-\rho ,t_j)\) belong to \({\Omega }^{N,M}_\ell \) and \(\Omega ^{N,M}_r,\) respectively. Therefore, Lemma 4 gives

$$\begin{aligned} ||{\overline{U}}_m-U^{[1]}_m||_{{\overline{\Omega }}^{N,M}_m}\le \mu _{\rho }+CN^{-2}. \end{aligned}$$
(19)

Combining (17), (18), and (19), we obtain

$$\begin{aligned} ||{\overline{U}}-U^{[1]}||_{{\overline{\Omega }}^{N,M}} \le \mu _{\rho }+CN^{-2}. \end{aligned}$$
(20)

Since \((\rho ,t_j)\in {\overline{\Omega }}^{N,M}_{\ell }\) and \((1-\rho ,t_j)\in {\overline{\Omega }}^{N,M}_r\), by Lemma 6 one gets \(\mu _{\rho }\le C(\Delta t + (N^{-1}\ln N)^2)\). Hence, using (20) and Lemma 6 in (11) the proof is complete. \(\square \)

In the next theorem, for \(\rho =1/4,\) we first obtain a bound on \(||{\overline{U}}-U^{[k]}||_{{\overline{\Omega }}^{N,M}}\) and then combine it with Lemma 6 to get a bound on \(||u-U^{[k]}||_{{\overline{\Omega }}^{N,M}}.\)

Theorem 2

Suppose \(U^{[k]}\) is the approximation of the exact solution u of (1)–(3), generated by the k-th iterate of the method. Then, for \(\rho =1/4,\) it holds

$$\begin{aligned} ||u-U^{[k]}||_{{\overline{\Omega }}^{N,M}}\le C\left( (5/6)^{k}+(\Delta t +(N^{-1}\ln N)^2)\right) . \end{aligned}$$
(21)

Proof

The following notation is used

$$\begin{aligned} \nu ^{[k]}= & {} \max \left\{ ||{\overline{U}}_{r}-U^{[k]}||_{{\overline{\Omega }}^{N,M}_{r}\backslash {\overline{\Omega }}_m}, ||{\overline{U}}_{\ell }-U^{[k]}||_{{\overline{\Omega }}^{N,M}_{\ell }\backslash {\overline{\Omega }}_m},||{\overline{U}}_{m}-U^{[k]}||_{{\overline{\Omega }}^{N,M}_{m}}\right\} ,\\ \mu _{2\rho }= & {} \displaystyle \max \left\{ \max _{t_j\in \Omega _t^M}|({\overline{U}}_{\ell }-{\overline{U}}_{m})(2\rho ,t_j)|, \max _{t_j\in \Omega _t^M}|({\overline{U}}_{r}-{\overline{U}}_{m})(1-2\rho ,t_j)| \right\} . \end{aligned}$$

We have

$$\begin{aligned} \left\{ \begin{array}{ll} L^{N,M}_\ell ( {\overline{U}}_\ell -U_\ell ^{[1]})=0 &{} \text{ in }~\Omega _\ell ^{N,M}, \\ ({\overline{U}}_\ell -U_\ell ^{[1]})(x_i,0)=0, &{} \text{ if }~x_i\in \overline{{\textsf{D}}}^{N}_{\ell },\\ \Gamma _\ell ^{N,M}({\overline{U}}_\ell -U_\ell ^{[1]})(0,t_j)=0 &{} \text{ if }~t_j\in \Omega _t^{M},\\ |({\overline{U}}_\ell -U_\ell ^{[1]})(2\rho ,t_j)|\le \mu ^{[1]} &{} \text{ if }~t_j\in \Omega _t^{M}.~~~~~~~~~~~~ \end{array} \right. \end{aligned}$$

We define

$$\begin{aligned}\Psi ^\pm (x_i,t_j)=\mu ^{[1]}\frac{x_i+\sqrt{\varepsilon }}{2\rho +\sqrt{\varepsilon }} \pm ({\overline{U}}_\ell -U_\ell ^{[1]})(x_i,t_j),\end{aligned}$$

which verify that \(\Psi ^\pm (x_i,0)\ge 0,\) \(x_i \in \overline{{\textsf{D}}}_\ell ^N;\) \(\Gamma _\ell ^{N,M}\Psi ^\pm (0,t_j)\ge 0, \Psi ^\pm (2\rho ,t_j)\ge 0,\) \(t_j\in \Omega _t^{M};\)

$$\begin{aligned} L^{N,M}_\ell \Psi ^\pm (x_i,t_j)=\left( \frac{x_i+\sqrt{\varepsilon }}{2\rho +\sqrt{\varepsilon }}\right) \mu ^{[1]}a(x_i,t_j)\pm 0\ge 0, (x_i,t_j)\in \Omega _\ell ^{N,M}. \end{aligned}$$

So, Lemma 4 gives

$$\begin{aligned} |({\overline{U}}_\ell -U_\ell ^{[1]})(x_i,t_j)|\le \mu ^{[1]}\left( \frac{x_i+\sqrt{\varepsilon }}{2\rho +\sqrt{\varepsilon }}\right) . \end{aligned}$$
(22)

Since \(x_i\le \rho \) for \((x_i,t_j)\in {{\overline{\Omega }}_\ell ^{N,M}\backslash {\overline{\Omega }}_m},\) it follows from (22) that

$$\begin{aligned} ||{\overline{U}}_\ell -U_\ell ^{[1]}||_{{\overline{\Omega }}_\ell ^{N,M}\backslash {\overline{\Omega }}_m} \le \frac{5}{6}\mu ^{[1]}. \end{aligned}$$
(23)

Similarly, we have

$$\begin{aligned} ||{\overline{U}}_r-U_r^{[1]}||_{{\overline{\Omega }}_r^{N,M}\backslash {\overline{\Omega }}_m} \le \frac{5}{6}\mu ^{[1]}. \end{aligned}$$
(24)

For \({\overline{U}}_m-U_m^{[1]},\) we have

$$\begin{aligned}L^{N,M}_m({\overline{U}}_m-U_m^{[1]})=0~~\text{ in }~ \Omega _m^{N,M},~~({\overline{U}}_m-U_m^{[1]})(x_i,0) =0,~\text{ if }~x_i \in \overline{{\textsf{D}}}_{m}^{N};\end{aligned}$$

and for \(t_j\in \Omega _t^M,\)

$$\begin{aligned}{} & {} |({\overline{U}}_m-U_m^{[1]})(\rho ,t_j)| =|({\overline{U}}_m-{\mathcal {I}}_jU_\ell ^{[1]})(\rho ,t_j)| \le \mu _{\rho }+\frac{5}{6}\mu ^{[1]}\\{} & {} \quad ~~~(\text{ since }~(\rho ,t_j)\in {\overline{\Omega }}^{N,M}_\ell )\\{} & {} |({\overline{U}}_m-U_m^{[1]})(1-\rho ,t_j)|=|({\overline{U}}_m-{\mathcal {I}}_jU_r^{[1]})(1-\rho ,t_j)|\le \mu _{\rho }+\frac{5}{6}\mu ^{[1]}\\{} & {} \quad ~~~(\text{ since }~(1-\rho ,t_j)\in {\overline{\Omega }}^{N,M}_r).\end{aligned}$$

Therefore, Lemma 4 gives

$$\begin{aligned} ||{\overline{U}}_m-U^{[1]}_m||_{{\overline{\Omega }}^{N,M}_m} \le \mu _{\rho }+\frac{5}{6}\mu ^{[1]}. \end{aligned}$$
(25)

We next obtain an estimate for \(\mu ^{[2]}.\) Since \((2\rho ,t_j)\) and \((1-2\rho ,t_j)\) belong to \({\overline{\Omega }}^{N,M}_m,\) it follows that

$$\begin{aligned} |({\overline{U}}_{\ell }-{\mathcal {I}}_jU^{[1]})(2\rho ,t_j)|\le & {} \mu _{2\rho }+\mu _{\rho }+\frac{5}{6}\mu ^{[1]}\\ \text{ and }~~~~~~~~|({\overline{U}}_{r}-{\mathcal {I}}_jU^{[1]})(1-2\rho ,t_j)|\le & {} \mu _{2\rho }+\mu _{\rho }+\frac{5}{6}\mu ^{[1]}. \end{aligned}$$

Thus, \(\mu ^{[2]} \le \mu _{2\rho }+\mu _{\rho }+\frac{5}{6}\mu ^{[1]}.\) Therefore, it holds that

$$\begin{aligned}\max \{\nu ^{[1]},\mu ^{[2]}\} \le \lambda + \frac{5}{6}\mu ^{[1]},~~~\lambda =\mu _{2\rho }+\mu _{\rho }.\end{aligned}$$

After repeating the above arguments one gets

$$\begin{aligned}\max \{\nu ^{[k]},\mu ^{[k+1]}\} \le \lambda +\frac{5}{6}\mu ^{[k]}.\end{aligned}$$

We simplify this to get

$$\begin{aligned}\mu ^{[k]} \le 6\lambda + \left( \frac{5}{6}\right) ^{k-1}\mu ^{[1]}.\end{aligned}$$

Thus, it is

$$\begin{aligned} \nu ^{[k]} \le 6\lambda +\left( \frac{5}{6}\right) ^{k}\mu ^{[1]}. \end{aligned}$$
(26)

Further, since \((2\rho ,t_j),\) \((1-2\rho ,t_j)\in {\overline{\Omega }}^{N,M}_m,\) \((\rho ,t_j)\in {\overline{\Omega }}^{N,M}_{\ell },\) and \((1-\rho ,t_j)\in {\overline{\Omega }}^{N,M}_{r},\) by Lemma 6\(\lambda \le C(\Delta t +(N^{-1}\ln N)^2).\) Since \(\mu ^{[1]}\le C,\) using (26) and Lemma 6 in (11) the proof is complete. \(\square \)

Fig. 1
figure 1

Solution plots for Example 1 taking \(N=64,~M=16,\) and various values of \(\varepsilon \)

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5 Numerical results

Numerical results considering a couple of examples are given that verify the convergence theory.

Example 1

Consider

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u(x,t)}{\partial t}-\varepsilon \frac{\partial ^2u(x,t)}{\partial x^2}+(1+xe^{-t})u(x,t)=f(x,t) &{} (x,t)\in \Omega ={\textsf{D}}\times (0,1], \\ u(0,t)-\sqrt{\varepsilon } \frac{\partial u}{\partial x}(0,t)=g_\ell (t) &{} t\in (0,1], \\ u(1,t)+\sqrt{\varepsilon } \frac{\partial u}{\partial x}(1,t)=g_r(t) &{} t\in (0,1], \\ u(x,0)=0 &{} x\in [0,1]. \end{array} \right. \end{aligned}$$

where \(f,~g_\ell ,\) and \(g_r\) are such that

$$\begin{aligned}u(x,t)=t\left[ \frac{e^{-x/\sqrt{\varepsilon }}+e^{(x-1)/\sqrt{\varepsilon }}}{1+e^{-1/\sqrt{\varepsilon }}}-\cos ^2({\pi x})\right] .\end{aligned}$$
Fig. 2
figure 2

Solution plots for Example 2 taking \(N=64,~M=16,\) and various values of \(\varepsilon \)

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Table 1 Errors and convergence orders for Example 1
Full size table
Table 2 Iterations for Example 1
Full size table

The solution plots for various values of \(\varepsilon \) are given in Fig. 1. Note that the boundary layers are close to \(x=0,1.\) Taking the tolerance error \(tol=N^{-2},\) we compute the approximate solution and denote it by \(U^{N,\Delta t}.\) We then evaluate parameter uniform errors and convergence orders as follows

$$\begin{aligned} E^{N,\Delta t}=\max _{\forall \varepsilon }E^{N,\Delta t}_\varepsilon ~~\text{ and }~~ R^{N,\Delta t}=\log _2\left( \frac{E^{N,\Delta t}}{E^{2N,\Delta t/4}}\right) ,\end{aligned}$$

where \(E^{N,\Delta t}_\varepsilon =||u-U^{N,\Delta t}||_{{\overline{\Omega }}^{N,M}}\) are the maximum pointwise errors. Table 1 displays the computed errors \(E^{N,\Delta t}_\varepsilon ,~ E^{N,\Delta t},\) and parameter uniform orders \( R^{N,\Delta t}\) for Example 1. Clearly, we can confirm the parameter uniform convergence from this table. In addition, Table 2 gives the iterations needed for convergence; from this, we observe that one iteration is enough to get the desired results for very small values of \(\varepsilon .\)

Example 2

Consider

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u(x,t)}{\partial t}-\varepsilon \frac{\partial ^2u(x,t)}{\partial x^2}+\frac{1+x^2}{2}u(x,t)=t^3 &{} (x,t)\in (0,1)\times (0,1], \\ u(0,t)-\sqrt{\varepsilon } \frac{\partial u}{\partial x}(0,t)=-(128/35)\pi ^{-1/2}t^{7/2} &{} t\in (0,1], \\ u(1,t)+\sqrt{\varepsilon } \frac{\partial u}{\partial x}(1,t)=-(128/35)\pi ^{-1/2}t^{7/2} &{} t\in (0,1], \\ u(x,0)=0 &{} x\in [0,1]. \end{array} \right. \end{aligned}$$

The solution plots for various values of \(\varepsilon \) are given in Fig. 2. Note that the boundary layers are close to \(x=0,1.\) Since the solution is unknown, we compute another approximate solution to calculate the uniform errors and uniform convergence orders as follows

$$\begin{aligned}{} & {} E^{N,\Delta t}_\varepsilon =\max _{(x_i,t_j)\in {\overline{\Omega }}^{N,M}}|U^{2N,\Delta t/4}(x_i,t_j)-U^{N,\Delta t}(x_i,t_j)|,~~ E^{N,\Delta t}=\max _{\varepsilon }E^{N,\Delta t}_\varepsilon ,\\{} & {} R^{N,\Delta t}=\log _2\left( \frac{E^{N,\Delta t}}{E^{2N,\Delta t/4}}\right) ,\end{aligned}$$

where \(U^{2N,\Delta t/4}\) is obtained taking in each subdomain \(2N +1\) points in space and \(\Delta t/4\) mesh size in time, but using the same subdomain parameter \(\rho \) as is considered for \(U^{N,\Delta t}.\) The error plots for various values of \(\varepsilon \) are given in Figs. 3 and 4 for Examples 1 and 2 respectively.

Table 3 Errors and convergence orders for Example 2
Full size table
Table 4 Iterations for Example 2
Full size table

The computed errors \(E^{N,\Delta t}_\varepsilon ,~ E^{N,\Delta t},\) and uniform convergence rates \( R^{N,\Delta t}\) for Example 2 are tabulated in Table 3, showing the uniform convergence of the method. Table 4 gives iterations that are performed to achieve the convergence for Example 2. One can see that only one iteration is needed to get the solution up to the desired accuracy for very small values \(\varepsilon \).

Fig. 3
figure 3

Error plots for Example 1 taking \(N=64,~M=16,\) and various values of \(\varepsilon \)

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Fig. 4
figure 4

Error plots for Example 2 taking \(N=64,~M=16,\) and various values of \(\varepsilon \)

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6 Conclusions

An SWR technique to solve singularly perturbed parabolic reaction-diffusion equations with Robin boundary conditions is developed and analyzed in this paper. The original domain is initially divided into three overlapping subdomains. The problem is discretized using the backward difference and central difference schemes for the time and space derivatives, respectively. It is shown that the proposed scheme is unconditionally stable. Error analysis is also discussed in this work, and it is demonstrated that the proposed approach is uniformly convergent with order almost two in space and one in time. Furthermore, it is shown that for small values of \(\varepsilon \), one iteration is sufficient to achieve the specified accuracy. The idea discussed in this paper can also be extended to singularly perturbed semilinear differential equations having boundary conditions of Robin type. Further, it is our intention in the future to extend the SWR technique to higher dimensional singularly perturbed problems.