Introduction

   Several partial differential equations found in practice are parameter-dependent, with a singularly perturbed nature for small values of this parameters. The solutions to these problems have boundary layers (and/or interior layers due to the discontinuities in the convection and source terms), which are near the boundary of the domain where the solution has an extremely high gradient. The layers might be either regular (exponential) or of parabolic type (characteristic). The problem considered is of the convection-reaction-diffusion type, which is one of the most common singular perturbation problems in the literature.

Singularly perturbed elliptic problems are widespread in mathematical modeling, ranging from simulation of oil and gas reservoirs, as well as magnetohydrodynamic flow, to chemical flow reactor theory [1], to boundary layers influenced by suction (or blowing) of some fluid and semiconductor model [2]. In most cases the equations are too complicated to be solved exactly, so the use of numerical techniques is required. On the other hand, classical techniques can utterly fail in the presence of layers (see, e.g. [3, 4]), thus this is a highly active and interesting topic of research for numerical analysts.

There is a lot of literature on parameter-uniform numerical methods (see, for example, [3, 5]).

Butuzov [6] studied the asymptotic structure of a solution to a problem similar to the one under consideration here, and stablished a substantial relationship between the ordering of small parameters. When \(\mu =O(\epsilon ^{1/2}),\) we are approaching the reaction-diffusion type with layer structure. The solutions to 2-D reaction-diffusion problems contain characteristic (parabolic) boundary layers along all edges of the unit square. Clavero considered a class of 2-D reaction-diffusion singularly perturbed problems in [7] and used a finite difference method with a piecewise uniform Shishkin mesh to find a more accurate estimate. When \(\mu =1\), we are approaching the convection-diffusion type with a regular (exponential) layer near the outflow of the boundary, and a corner layer near the boundary layers junction. In a 2-D singularly perturbed convection-diffusion problem, the solution has boundary and interior layers in the domain. Therefore, the derivative’s norms of z in [8] show that the solution z contains regular (exponential) boundary layers along all of the unit square’s edges, while certain bounds in [9] demonstrate both exponential and parabolic boundary layers in z. In [10], they introduced the asymptotic character of singularly perturbed convection-diffusion problems to better understand their solution and related challenges with numerical techniques. Lin et al. [11] analyzed the weak form of 2-D singularly perturbed convection-diffusion problems using an effective approach based on local discontinuous Galerkin (LDG) discretization. Nhan and Vulanovic [12] explain a class of 2-D convection-diffusion singularly perturbed problems using a complete finite-difference scheme with a Bakhvalov mesh to find more accurate estimates. When \(\mu\) moves away from \(\epsilon ^ {1/2}\) and remains small when compared to \(\mu =1\), we have an entirely different layer structure.

In a 2-D singularly perturbed convection-diffusion problem with two parameters, Zhang and Lv [13] consider an efficient approach based on the finite-element method using a Bakhvalov mesh. In O’Riordan et al. [14, 15], they consider the finite-difference method with a piecewise uniform Shishkin mesh for a two-parameter convection-diffusion singularly perturbed 2-D problems. Teofanov and Roos [16, 17] handle an efficient approach based on the finite-element method using linear or bilinear elements in a piecewise uniform Shishkin mesh for a 2-D singularly perturbed convection-diffusion problem with two parameters. In that paper, they discuss the decomposition of the solution and derivative’s bounds on the components. In this study, we develop a finite-difference method (FDM) for solving two-parameter singularly perturbed convection-diffusion 2-D problems, where a, b and f are discontinuous along two lines, namely \(x=d\) and \(y=d\).

Let us consider a two-parameter singularly perturbed 2-D steady-state convection-reaction-diffusion equation

$$\begin{aligned} L_{\epsilon ,\mu }z(x,y) = f(x,y), \quad \forall (x,y)\in \Omega , \qquad z(x,y) = g(x,y), \quad \forall (x,y) \in {\partial \Omega }, \end{aligned}$$
(1.1a)

where the differential operator is represented by

$$\begin{aligned} L_{\epsilon ,\mu }z(x,y) \equiv \epsilon ^2\bigg (\dfrac{\partial ^2 z}{\partial x^2} + \dfrac{\partial ^2 z}{\partial y^2}\bigg )+\mu ^2\bigg (a(x,y)\dfrac{\partial z}{\partial x}+b(x,y)\dfrac{\partial z}{\partial y}\bigg ) - c(x,y) z. \end{aligned}$$
(1.1b)

The two small perturbation parameters satisfy \(0<\epsilon ,\mu<<1\). The problem’s domain is \(\Omega =\bigcup \limits _{k=1}^{4}\Omega _k\), being \(\Omega _1=(0,d)\times (0,d),\) \(\Omega _2=(d,1)\times (0,d),\) \(\Omega _3=(0,d)\times (d,1)\) and \(\Omega _4=(d,1)\times (d,1).\) Let \(\bar{\Omega }=[0,1]\times [0,1]\) and \(\Gamma _1=\{(d,y):0\le y \le 1\},\Gamma _2=\{(x,d):0\le x \le 1\},\) with d any point in (0, 1).

We assume that the convection terms are bounded as follows with

$$\begin{aligned}{} & {} \alpha _1^*>a(x,y)\ge \alpha _1>0,\,\, \text {for} \,\, x<d, \,\, -\alpha _2^*< a(x,y)\le -\alpha _2<0,\,\,\text {for}\,\, x>d,\\{} & {} \beta _1^*>b(x,y)\ge \beta _1>0,\,\, \text {for} \,\, y<d, \,\, -\beta _2^*< b(x,y)\le -\beta _2 <0,\,\,\text {for}\,\, y>d,\\{} & {} \alpha =\min (\alpha _1,\alpha _2),\, \alpha ^*=\max (\alpha _1^*,\alpha _2^*),\, \lambda =\min \limits _{\bar{\Omega }}\bigg \{\frac{c}{2a},\frac{c}{2b}\bigg \},\\{} & {} \beta =\min (\beta _1,\beta _2),\, \beta ^*=\max (\beta _1^*,\beta _2^*), \end{aligned}$$

for some constants \(\alpha _1,\,\alpha _2\) and \(\beta\), while the reaction coefficient verifies \(c(x,y) >0\). We will also assume that \(a|_{\Omega _k},\, b|_{\Omega _k},\,f|_{\Omega _k} \in C^{3,\gamma }({ \Omega _k})\), \(c \in C^{3,\gamma }(\bar{\Omega })\), and \(g \in C^{4,\gamma }({\Omega })\), for some \(\gamma \in (0,1],\, k=1,2,3,4\). Further, we assume that the exact solution is such that \(z \in C^{4,\gamma }({ \Omega _k})\).

In our problem (1.1), the source term f(x, y) has a jump discontinuity at both lines \(x=d\) and \(y=d\), and the convection terms a(x, y), b(x, y),  have jump discontinuities at the line \(x=d\) or \(y=d\), respectively. So, it is congruent to denote the jump discontinuity in any function \(\kappa\) at a point \((x,y) \in \Omega\) along the lines parallel to the x- and y-axe as \([\kappa ](d,y)=\kappa (d^+,y) - \kappa (d^-,y)\) and \([\kappa ](x,d)=\kappa (x,d^+) - \kappa (x,d^-)\), respectively. The following symbols are used to specify the boundaries:

$$\begin{aligned}\Lambda _{1}= & {} \big \{(0,y)\,|\,(0 \le y< d) \cup (d< y \le 1 \big \}, \quad \Lambda _{2}=\big \{(x,0)\,|\,(0\le x< d)\cup (d< x \le 1)\big \},\\ \Lambda _{3}= & {} \big \{(1,y)\,|\,(0 \le y< d)\cup (d< y \le 1)\big \},\quad \Lambda _{4}=\big \{(x,1)\,|\,(0 \le x< d) \cup (d < x \le 1)\big \}, \end{aligned}$$

and \(\Lambda =\Lambda _1\cup \Lambda _2\cup \Lambda _3\cup \Lambda _4.\) Recalling from (1.1) that \(z=g\) on the boundary, we denote by \(g_i\) the restriction of g onto \(\Lambda _i\), \(i=1,2,3,4\).

We further denote the continuous subsets of the boundaries, and the interior line segments on both sides of the discontinuities as \(\Lambda _{k,j},\, c_{k,j}\) where \(j=1,2,3,4\) (which indicate the edges and corners of \(\Omega _k\) respectively) (see Fig. 1).

Fig. 1
figure 1

Notation for Subregions and Domain Boundaries

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The article is structured as follows. In Section 2, we derived the minimum principle, a stability result and bounds of the exact solution and its derivatives exhibiting their dependence on the singular perturbation parameters. Section 3 explores the numerical approach of the standard 5-point finite difference scheme built on a Shishkin mesh. In Section 4, we derived the error estimation, which results in an almost first-order convergence. At the end of Section 5, some test problems are provided to verify the theoretical results.

A Priori Bounds of the Solution and its Derivatives

The present section contains the minimum principle, a stability result, and some useful bounds for the derivatives of the true solution. In addition, we obtain some bounds of the regular, singular, and corner components of the solution.

Lemma 2.1

(Minimum principle) Let \(L_{\epsilon ,\mu }\) be the differential operator given in (1.1). If \(\phi (x,y)\ge 0\) on \(\partial \Omega\),  \(L_{\epsilon ,\mu }\phi (x,y)\le 0\) for all \((x,y)\in \Omega\), \(\big [\frac{\partial \phi }{\partial x}\big ](d,y)\le 0\) on \(\Gamma _1\) and \(\big [\frac{\partial \phi }{\partial y}\big ](x,d)\le 0\) on \(\Gamma _{2}\) then it is \(\phi (x,y)\ge 0\) for all \((x,y)\in \bar{\Omega }.\)

Proof

Consider the function \(\omega\) on \(\bar{\Omega }\) defined through \(\phi (x,y)=\omega (x,y)\psi (x,y)\), with the function

$$\begin{aligned}\psi (x,y)=\exp \bigg (\frac{\mu ^2(d-x)\lambda _1(x)}{2\epsilon ^2}+\frac{\mu ^2(d-y)\lambda _2(y)}{2\epsilon ^2}\bigg ),\quad (x,y)\in \bar{\Omega }.\end{aligned}$$

The functions \(\lambda _1(x),\,\lambda _2(y)\) on [0, 1] are defined as

$$\begin{aligned} \lambda _1(x)= {\left\{ \begin{array}{ll} \lambda _{11}, &{}\, \text {if}\, x\le d,\\ -\lambda _{12}, &{}\, \text {if}\, x>d, \end{array}\right. } \qquad \lambda _2(y)= {\left\{ \begin{array}{ll} \lambda _{21}, &{}\, \text {if}\, y\le d,\\ -\lambda _{22}, &{}\, \text {if}\, y>d, \end{array}\right. } \end{aligned}$$

where \(\alpha \ge \lambda _{11}>\lambda _{12}>0,\) and \(\beta \ge \lambda _{21}>\lambda _{22}>0\) are some constants. Let be \(\omega (x^*,y^*)=\min \limits _{(x,y)\in \bar{\Omega }}\{\omega (x,y)\}\). If \(\omega (x^*,y^*)\ge 0\), there is nothing to prove. Suppose \(\omega (x^*,y^*)<0.\) Then, by the assumption on the boundary values, either the point \((x^*,y^*)\in \Omega\) or \((x^*,y^*) \in \Gamma _1\cup \Gamma _2\). Let us consider both cases.

Case(i): Firstly, assume that \((x^*,y^*)\in \Omega\).

Then, at the point \((x^*,y^*)\) it is \(\dfrac{\partial \omega }{\partial x}{(x^*,y^*)}=\dfrac{\partial \omega }{\partial y}{(x^*,y^*)}=0\) and \(\dfrac{\partial ^{2} \omega }{\partial x^{2}}{(x^*,y^*)}\ge 0, \dfrac{\partial ^{2} \omega }{\partial y^{2}}{(x^*,y^*)}\ge 0\), we have

$$\begin{aligned} L_{\epsilon ,\mu }\phi (x^*,y^*)&= \psi (x^*,y^*)\Bigg (\epsilon ^2\Delta \omega +\Bigg (\frac{\mu ^4\lambda _1(x^*)}{2\epsilon ^2}\Big (\frac{\lambda _1(x^*)}{2}- a(x^*,y^*)\Big )\\&\quad +\frac{\mu ^4\lambda _2(y^*)}{2\epsilon ^2}\Big (\frac{\lambda _2(y^*)}{2}- b(x^*,y^*)\Big )\Bigg )\omega (x^*,y^*)\\&\quad-c\omega (x^*,y^*) \Bigg )>0, \end{aligned}$$

which contradicts the hypothesis.

Case(ii): Now suppose that \((x^*,y^*) \in \Gamma _1\cup \Gamma _2.\)

Here, either \((x^*,y^*) = (d,y^*),\) or \((x^*,y^*) = (x^*,d)\). Let us assume \((x^*,y^*) = (d,y^*)\). Since \(\omega\) takes its minimum value at \((x^*,y^*)\), this implies that \(\frac{\partial \omega }{\partial x}(d^+,y^*) \ge 0\) and \(\frac{\partial \omega }{\partial x}(d^-,y^*) \le 0.\) Then, it is evident that \([\frac{\partial \omega }{\partial x}](d,y^*) \ge 0\). Then, since \(\omega (d,y^*)<0,\) it follows that

$$\begin{aligned} \bigg [\frac{\partial \phi }{\partial x}\bigg ](d,y^*)=\exp \bigg (\frac{\mu ^2\lambda _2(d-y^*)}{2\epsilon ^2}\bigg )\bigg (\bigg [\frac{\partial \omega }{\partial x}\bigg ](d,y^*)+\mu ^2\frac{\lambda _{12}-\lambda _{11}}{2\epsilon ^2}\omega (d,y^*)\bigg )>0, \end{aligned}$$

which contradicts the hypothesis \([\frac{\partial \phi }{\partial x}](x,y)\le 0,\,\,\forall (x,y)\in \Gamma _1.\) The case when \((x^*,y^*) = (x^*,d)\) can be proved similarly. This completes the proof. \(\square\)

A consequence of this minimum principle is the parameter uniform boundedness of the solution of (1.1) given below.

Lemma 2.2

(Stability result) Let z(x, y) be the solution of (1.1). Then, it holds

$$\begin{aligned} ||z(x,y)||_{\bar{\Omega }} \le \dfrac{1}{\nu } ||L_{\epsilon ,\mu }z||_{\bar{\Omega }}+\max \bigg \{ || z ||_{\Lambda _{1}},|| z ||_{\Lambda _{2}},|| z ||_{\Lambda _{3}},|| z ||_{\Lambda _{4}}\bigg \}, \end{aligned}$$

where ||.|| represents the pointwise maximum norm and \(\nu =\min \{\alpha ,\beta \}\).

Proof

We define the barrier functions

$$\begin{aligned} \phi ^{\pm }(x,y)= {\left\{ \begin{array}{ll} M+\frac{||L_{\epsilon ,\mu }z||_{\bar{\Omega }}}{\nu }\bigg (\frac{1}{2}+\frac{x}{8}+\frac{y}{8}-\frac{d}{4}\bigg )\pm z(x,y), \quad \, \, (x,y)\in [0, d]\times [0,d],\\ M+\frac{||L_{\epsilon ,\mu }z||_{\bar{\Omega }}}{\nu }\bigg (\frac{1}{2}-\frac{x}{4}+\frac{y}{8}+\frac{d}{8}\bigg )\pm z(x,y), \quad (x,y)\in (d,1]\times [0,d],\\ M+\frac{||L_{\epsilon ,\mu }z||_{\bar{\Omega }}}{\nu }\bigg (\frac{1}{2}+\frac{x}{8}-\frac{y}{4}+\frac{d}{8}\bigg )\pm z(x,y), \quad (x,y)\in [0,d]\times (d,1],\\ M+\frac{||L_{\epsilon ,\mu }z||_{\bar{\Omega }}}{\nu }\bigg (\frac{1}{2}-\frac{x}{4}-\frac{y}{4}+\frac{d}{2}\bigg )\pm z(x,y), \quad (x,y)\in (d,1]\times (d,1], \end{array}\right. } \end{aligned}$$

where \(M = \max \{ || z ||_{\Lambda _{1}},|| z ||_{\Lambda _{2}},|| z ||_{\Lambda _{3}},|| z ||_{\Lambda _{4}}\}\).

Then, clearly \(\phi ^{\pm }(x,0)\ge 0,\, \phi ^{\pm }(0,y)\ge 0,\, \phi ^{\pm }(x,1)\ge 0, \,\phi ^{\pm }(1,y)\ge 0\). For each \((x,y)\in \Omega ,\) we have

$$\begin{aligned} L_{\epsilon ,\mu }\phi ^{\pm }(x,y)\le 0. \end{aligned}$$

Since \(z(x,y)\in C^0\bar{(\Omega )}\cap C^{2}(\Omega ),\) we have

$$\begin{aligned} \bigg [\dfrac{\partial \phi ^{\pm }}{\partial x}\bigg ](d,y)=\frac{-3||L_{\epsilon ,\mu }z||_{\bar{\Omega }}}{8\nu }\pm \bigg [\dfrac{\partial z^{\pm }}{\partial x}\bigg ](d,y)\le 0,\\ \bigg [\dfrac{\partial \phi ^{\pm }}{\partial y}\bigg ](x,d)=\frac{-3||L_{\epsilon ,\mu }z||_{\bar{\Omega }}}{8\nu }\pm \bigg [\dfrac{\partial z^{\pm }}{\partial y}\bigg ](x,d)\le 0. \end{aligned}$$

It follows from Lemma 2.1 that \(\phi ^{\pm }(x,y)\ge 0\),   \(\forall (x,y) \,\in \bar{\Omega },\) which allows to get the bound on \(||z(x,y)||_{\bar{\Omega }}\). \(\square\)

The derivatives of the solution satisfy the parameter-explicit bounds shown as follows.

Lemma 2.3

Let z be the solution of (1.1). Then, for \(1\le i+j \le 4\), it holds

$$\begin{aligned}&\bigg |\bigg |\dfrac{\partial ^{i+j} z}{\partial x^{i} \partial y^{j}}\bigg |\bigg |_{\Omega _k}\le C\epsilon ^{-(i+j)},\, \text {if}\,\, \alpha \mu ^2\le \lambda \epsilon , \end{aligned}$$
(2.1)
$$\begin{aligned}&\bigg |\bigg |\dfrac{\partial ^{i+j} z}{\partial x^{i} \partial y^{j}}\bigg |\bigg |_{\Omega _k}\le C\bigg (\frac{\epsilon }{\mu }\bigg )^{-(2i+2j)},\, \text {if}\,\, \alpha \mu ^2> \lambda \epsilon , \end{aligned}$$
(2.2)

where C is independent of \(\epsilon\) and \(\mu\).

Proof

It can be easily obtained using standard procedures, as in [18, 19]. \(\square\)

Now, we decompose the solution z(x, y) into the regular and singular components. The regular components \(r_k(x,y),\, k=1,\,2, \,3, \, 4,\) are obtained, respectively, as the solution of the following problems

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{\epsilon ,\mu } r_k(x,y) = f(x,y), \,\, \forall (x,y) \in \Omega _k,\,k=1,2,3,4\\ r_k(x,y)=g_1(y),\quad \forall (x,y)\in \Lambda _{k,1},\,k=1,3,\quad r_k(x,y)=g_2(x),\quad \forall (x,y)\in \Lambda _{k,2},\,k=1,2,\\ r_k(x,y)=g_3(y),\quad \forall (x,y)\in \Lambda _{k,3},\,k=2,4,\quad r_k(x,y)=g_4(x),\quad \forall (x,y)\in \Lambda _{k,4},\,k=3,4, \\ {[}r_k{]}(x,y)=0,\quad \forall (x,y)\in \Gamma _1\cup \Gamma _2,\,k=1,2,3,4,\\ {[}(r_k)_x{]}(x,y)=0,\quad \forall (x,y)\in \Gamma _1,\, \quad [(r_k)_y](x,y)=0,\quad \forall (x,y)\in \Gamma _2,\,k=1,2,3,4. \end{array}\right. } \end{aligned}$$
(2.3)

Lemma 2.4

The regular components \(r_k(x,y)\) at (2.3) and their derivatives satisfy the bounds:

$$\begin{aligned} \bigg |\bigg |\frac{\partial ^{i+j} r_k}{\partial x^i\partial y^j}\bigg |\bigg |_\Omega \le C\bigg (1+{\epsilon }^{2-(i+j)}\bigg ),\qquad&\text {for} \quad 1\le i+j \le 4,\quad \text {if} \quad \alpha \mu ^2\le \lambda \epsilon ,\\ \bigg |\bigg |\frac{\partial ^{i+j} r_k}{\partial x^i\partial y^j}\bigg |\bigg |_\Omega \le C\bigg (1+\bigg (\frac{\epsilon }{\mu }\bigg )^{4-(2i+2j)}\bigg ),\quad&\text {for} \quad 1\le i+j \le 4,\quad \text {if} \quad \alpha \mu ^2> \lambda \epsilon . \end{aligned}$$

Proof

Let us consider the two cases.

Case(i): Firstly, assume that \(\alpha \mu ^2\le \lambda \epsilon\).

Suppose the regular component \(r_k(x,y),\, k=1,\,2, \,3, \, 4\) can be decomposed as

$$r_k(x,y)=r_{k,0}(x,y,\epsilon ,\mu )+\epsilon r_{k,1}(x,y,\epsilon ,\mu )+\epsilon ^2r_{k,2}(x,y,\epsilon ,\mu ), $$
(2.4)

where \(r_{k,0}\), \(r_{k,1}\) and \(r_{k,2}\) are the solutions to the following problems without any compatibility inequalities:

$$\begin{aligned} -r_{k,0}&=f,\,\,\,\,r_{k,1}=\epsilon \bigg (\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}\bigg ) r_{k,0}+\frac{\mu ^2}{\epsilon }\bigg (a\frac{\partial }{\partial x} +b \frac{\partial }{\partial y}\bigg )r_{k,0},\\ L_{\epsilon ,\mu }r_{k,2}&=-\epsilon \bigg (\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}\bigg ) r_{k,1}-\frac{\mu ^2}{\epsilon }\bigg (a\frac{\partial }{\partial x} +b \frac{\partial }{\partial y}\bigg )r_{k,1}, \\ {}&\quad r_{k,2} = 0, \,\, \forall (x,y) \in \Lambda _{k,j},\,k,j=1,\,2,\,3,\,4. \end{aligned}$$

Since, \(r_{k,0}\in C^{4,\gamma }(\bar{\Omega }_k),\, k=1,\,2,\,3,\,4\), we get \(\bigg (\dfrac{\partial ^2}{\partial x^2} + \dfrac{\partial ^2}{\partial y^2}\bigg ) r_{k,0}\in C^{2,\gamma }(\bar{\Omega }_k),\, k=1,\,2,\,3,\,4.\)

Applying Lemma 2.2 and Lemma 2.3 to the problem (2.3), it results that \(r_k\, \in \,C^{4,\gamma }({\Omega _k})\) and

$$\begin{aligned} \bigg |\bigg |\dfrac{\partial ^{i+j} r_k}{\partial x^i \partial y^j}\bigg |\bigg |\le C(1+\epsilon ^{2-(i+j)}), \quad 1\le i+j \le 4,\,\, k=1,\,2,\,3,\,4. \end{aligned}$$
(2.5)

Case(ii): \(\alpha \mu ^2 > \lambda \epsilon\).

Suppose the regular component \(r_k(x,y;\epsilon ,\mu ),\, k=1,\,2, \,3, \, 4\) can be decomposed as

$$\begin{aligned} r_k(x,y;\epsilon ,\mu )=r_{k,0}(x,y;\mu )+\epsilon ^2 r_{k,1}(x,y;\mu )+\epsilon ^4r_{k,2}(x,y;\epsilon ,\mu ), \end{aligned}$$

where

$$\begin{aligned} L_{\mu }r_{k,0}=f,\, r_{k,0}(x,y)=z(x,y),\, \forall (x,y)\in \Lambda _{3}\cup \Lambda _{4},\\ L_{\mu}r_{k,1}=-\bigg (\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}\bigg ) r_{k,0},\, r_{k,1}(x,y)=0,\, \forall (x,y)\in \Lambda _{3}\cup \Lambda _{4},\\ L_{\epsilon ,\mu }r_{k,2}=-\bigg (\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}\bigg ) r_{k,1}, \quad r_{k,2}(x,y) = 0, \,\, \forall (x,y) \in \Lambda _{k,j},\,k,j=1,\,2,\,3,\,4. \end{aligned}$$

Applying Lemma 2.6 and Lemma 2.3 to the problem (2.3) we get

$$\begin{aligned} \bigg |\bigg |\dfrac{\partial ^{i+j} r_k}{\partial x^i \partial y^j}\bigg |\bigg |\le C\bigg (1+\bigg (\frac{\epsilon }{\mu }\bigg )^{4-(2i+2j)}\bigg ), \quad 1\le i+j \le 4,\,\, k=1,\,2,\,3,\,4. \end{aligned}$$
(2.6)

\(\square\)

Now, let us consider the first order IBVP:

$$\begin{aligned} L_{\mu }z(x,y)=\mu ^2\bigg (a(x,y)\dfrac{\partial z}{\partial x}+b(x,y)\dfrac{\partial z}{\partial y}\bigg ) - c(x,y)z=f(x,y),\, \forall (x,y)\in \Omega , \end{aligned}$$
(2.7a)
$$\begin{aligned} z(x,y)=g_i,\,\, (x,y)\in \Lambda _{i},\,i=3,4. \end{aligned}$$
(2.7b)

Note that \(L_\mu\) satisfies the following comparison principle:

Lemma 2.5

Let \(L_{\mu }\) be the differential operator given in (2.7). If \(\phi (x,y)\ge 0\) on \(\Lambda _i,\, i=3,4\),  \(L_{\mu }\phi (x,y)\le 0\) for all \((x,y)\in \Omega\), \(\big [\frac{\partial \phi }{\partial x}\big ](d,y)\le 0\) on \(\Gamma _{1}\), and \(\big [\frac{\partial \phi }{\partial y}\big ](x,d)\le 0\) on \(\Gamma _{2}\) then it is \(\phi (x,y)\ge 0\) for all \((x,y)\in \bar{\Omega }.\)

Proof

The proof is similar to the one of Lemma 2.1. \(\square\)

Lemma 2.6

Let z(x, y) be the solution of problem (2.7). Then, it holds the stability estimate

$$\begin{aligned} ||z(x,y)||_{\bar{\Omega }} \le \dfrac{1}{\nu } ||L_{\mu }z||_{\Omega }+\max \bigg \{ || z ||_{\Lambda _3}, || z ||_{\Lambda _4}\bigg \}. \end{aligned}$$

Proof

This lemma can be proved similarly to Lemma 2.2. \(\square\)

Corresponding to the edge \(x=0\) in \(\Omega _1\) (see, Fig. 2), a layer function \(s_{1}\) exists that is determined by:

$$\begin{aligned} L_{\epsilon ,\mu }s_{1}(x,y)=0,\quad&\forall (x,y)\in \Omega , \end{aligned}$$
(2.8a)
$$\begin{aligned} s_{1}(x,y)= (z - r_1)(x,y), \quad&\forall (x,y) \in \Lambda _{1,1}, \end{aligned}$$
(2.8b)
$$\begin{aligned} s_{1}(x,y)=0, \quad&\forall (x,y) \in \Lambda _{3,1}\cup \Lambda _{2}\cup \Lambda _{3}\cup \Lambda _{4}, \end{aligned}$$
(2.8c)
$$\begin{aligned} s_1(x,y)=0,\quad&\forall (x,y)\in \Gamma _1\cup \Gamma _2, \end{aligned}$$
(2.8d)
$$\begin{aligned} \left[(s_1)_x\right](x,y)=0, \quad \forall (x,y)\in \Gamma _1,\, [({s}_{1})_y](x,y)=0, \quad&\forall (x,y)\in \Gamma _2. \end{aligned}$$
(2.8e)

The following lemmas give some bounds on the derivatives of the layer components, which are necessary for the convergence analysis.

Lemma 2.7

Let \(s_{1}\) be the boundary layer component satisfying the equations in (2.8). If \(\alpha \mu ^2\le \lambda \epsilon ,\) then it holds

$$\begin{aligned} |s_{1}(x,y)|\le & {} C \exp \bigg (-\frac{\alpha \lambda }{\epsilon }x\bigg ),\quad \\ \bigg |\bigg | \frac{\partial ^j s_{1}}{\partial y^j}\bigg |\bigg |\le & {} C(1+\epsilon ^{1-j}),\quad j=1,2,3,4. \end{aligned}$$

If \(\alpha \mu ^2> \lambda \epsilon ,\) then it holds

$$\begin{aligned} |s_{1}(x,y)|\le & {} C \exp \bigg (-\frac{\alpha \mu ^2}{\epsilon ^2}x\bigg ),\quad \\ \bigg |\bigg | \frac{\partial ^j s_{1}}{\partial y^j}\bigg |\bigg |\le & {} C \bigg (1+\bigg (\frac{\epsilon }{\mu }\bigg )^{-2j}\bigg ),\quad j=1,2,3,4. \end{aligned}$$
Fig. 2
figure 2

Situation of the layer functions in the domain \(\Omega\), when \(\alpha \mu ^2\le \lambda \epsilon\) and \(\alpha \mu ^2> \lambda \epsilon\)

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Proof

It can be referred from the works by O’Riordan et al. [14, 15]. \(\square\)

Similarly, as has been done for \(s_{1},\) for the different edges of \(\Omega _k,\, k=1,2,3,4,\) (see Fig. 1) we can consider the corresponding boundary layer components \(s_{i},\,\, i=2,3,\dots,16,\) (see Fig. 2) for which we can obtain similar bounds as in Lemma 2.7.

Related to the corner at \(c_{1,1}=(0,0)\) in \(\Omega _1\), we consider the corner layer component \(p_{1}\), which is determined by

$$\begin{aligned} L_{\epsilon ,\mu }p_{1}(x,y)=0,\quad&\forall (x,y) \in \Omega , \end{aligned}$$
(2.9a)
$$\begin{aligned} p_{1}(x,y) = -s_{1}(x,y), \,\, \forall (x,y) \in \Lambda _{1,1}, \quad p_{1}(x,y) = -s_{2}(x,y), \,\,&\forall (x,y) \in \Lambda _{1,2} , \end{aligned}$$
(2.9b)
$$\begin{aligned} p_{1}(x,y) = 0, \,\,&\forall (x,y) \in \Lambda _{2,2}\cup \Lambda _{3,1}\cup \Lambda _{3}\cup \Lambda _{4}, \end{aligned}$$
(2.9c)
$$\begin{aligned} p_1(x,y)=0,\quad&\forall (x,y)\in \Gamma _1\cup \Gamma _2, \end{aligned}$$
(2.9d)
$$\begin{aligned} \left[(p_1)_x\right](x,y)=0, \quad \forall (x,y)\in \Gamma _1,\, [({p}_{1})_y](x,y)=0, \quad&\forall (x,y)\in \Gamma _2. \end{aligned}$$
(2.9e)

Lemma 2.8

Let \(p_{1}\) be the corner layer component satisfying the equations in (2.9). If \(\alpha \mu ^2\le \lambda \epsilon ,\) then it holds

$$\begin{aligned} |p_{1}(x,y)|\le C \exp \bigg (-\frac{\alpha \lambda }{\epsilon }x\bigg )\exp \bigg (-\frac{\alpha \lambda }{\epsilon }y\bigg ),\quad \bigg |\bigg | \frac{\partial ^{i+j} p_{1}}{\partial x^{i}\partial y^{j}}\bigg |\bigg |\le C(1+\epsilon ^{-(i+j)}),\quad 1\le i+j\le 4. \end{aligned}$$

If \(\alpha \mu ^2> \lambda \epsilon ,\) then it holds

$$\begin{aligned} |p_{1}(x,y)|\le C \exp \bigg (-\frac{\alpha \mu ^2}{\epsilon ^2}x\bigg )\exp \bigg (-\frac{\alpha \mu ^2}{\epsilon ^2}y\bigg ),\quad \bigg |\bigg | \frac{\partial ^{i+j} p_{1}}{\partial x^i\partial y^j}\bigg |\bigg |\le C \bigg (\frac{\epsilon }{\mu }\bigg )^{-(2i+2j)},\quad 1\le i+j\le 4. \end{aligned}$$

Proof

It can be referred from the works by O’Riordan et al. [14, 15]. \(\square\)

Similarly, we can describe other corner layer components \(p_{k},\,k=2,3,\dots,16,\) corresponding to the different corners of \(\Omega _k,\,k=1,2,3,4\), which verify similar bounds as the ones in Lemma 2.8.

Finally, from the above lemmas we can establish the following theorem.

Theorem 2.9

The solution z of (1.1) may be written as

$$\begin{aligned} z=\sum _{k=1}^{4}r_{k}+\sum _{j=1}^{16}{s}_{j}+\sum _{j=1}^{16}{p}_{j}, \end{aligned}$$

where

$$\begin{aligned}\quad L_{\epsilon ,\mu }r_k = f, \quad L_{\epsilon ,\mu }{s}_{j}=0,\quad L_{\epsilon ,\mu }{p}_{j}=0,\,\, k=1,2,3,4,\,\,j=1,2,3,\dots,16.\end{aligned}$$

Furthermore, the regular and singular components and their derivatives satisfy the following bounds

\({\left\{ \begin{array}{ll} \bigg |\bigg |\dfrac{\partial ^{i+j} r_k}{\partial x^i \partial y^j}\bigg |\bigg |\le C(1+\epsilon ^{2-(i+j)}),\quad &{} 1\le i+j\le 4,\, k=1,2,3,4,\,\,\, \text {if}\,\, \alpha \mu ^2\le \lambda \epsilon ,\\ \bigg |\bigg |\dfrac{\partial ^{i+j} r_k}{\partial x^i \partial y^j}\bigg |\bigg |\le C(1+\big (\frac{\epsilon }{\mu }\big )^{4-(2i+2j)}),\quad &{} 1\le i+j\le 4,\, k=1,2,3,4,\,\,\, \text {if}\,\,\alpha \mu ^2> \lambda \epsilon ,\ \end{array}\right. }\)

$$\begin{aligned} {\left\{ \begin{array}{ll} |s_{1}(x,y)| &{}\le C e^{-\alpha \theta _1 x};\\ |s_{2}(x,y)| &{}\le C e^{-\beta \theta _1 y};\\ |s_{3}(x,y)| &{}\le C e^{-\alpha \theta _2 (d-x)};\\ |s_{4}(x,y)| &{}\le C e^{-\beta \theta _2 (d-y)};\\ \end{array}\right. }\\ {\left\{ \begin{array}{ll} |s_{5}(x,y)| &{}\le C e^{-\alpha \theta _2 (x-d)};\\ |s_{6}(x,y)| &{}\le C e^{-\beta \theta _1 y};\\ |s_{7}(x,y)| &{}\le C e^{-\alpha \theta _1 (1-x)};\\ |s_{8}(x,y)| &{}\le C e^{-\beta \theta _2 (d-y)};\\ \end{array}\right. }\\ {\left\{ \begin{array}{ll} |s_{9}(x,y)| &{}\le C e^{-\alpha \theta _1 x};\\ |s_{{10}}(x,y)| &{}\le C e^{-\beta \theta _2 (y-d)};\\ |s_{{11}}(x,y)| &{}\le C e^{-\alpha \theta _2 (d-x)};\\ |s_{{12}}(x,y)| &{}\le C e^{-\beta \theta _1 (1-y)};\\ \end{array}\right. }\\ {\left\{ \begin{array}{ll} |s_{{13}}(x,y)| &{}\le C e^{-\alpha \theta _2 (x-d)};\\ |s_{{14}}(x,y)| &{}\le C e^{-\beta \theta _2 (y-d)};\\ |s_{{15}}(x,y)| &{}\le C e^{-\alpha \theta _1 (1-x)};\\ |s_{{16}}(x,y)| &{}\le C e^{-\beta \theta _1 (1-y)};\\ \end{array}\right. } \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} |p_{{1}}(x,y)| &{}\le Ce^{-\alpha \theta _1x}e^{-\beta \theta _1y},\\ |p_{{2}}(x,y)| &{}\le Ce^{-\alpha \theta _2(d-x)}e^{-\beta \theta _1y},\\ |p_{{3}}(x,y)| &{}\le Ce^{-\alpha \theta _2(d-x)}e^{-\beta \theta _2(d-y)},\\ |p_{{4}}(x,y)| &{}\le Ce^{-\alpha \theta _1x}e^{-\beta \theta _2(d-y)},\\ \end{array}\right. }\\ {\left\{ \begin{array}{ll} |p_{{5}}(x,y)| &{}\le Ce^{-\alpha \theta _2(x-d)}e^{-\beta \theta _1y},\\ |p_{{6}}(x,y)| &{}\le Ce^{-\alpha \theta _1(1-x)}e^{-\beta \theta _1y},\\ |p_{{7}}(x,y)| &{}\le Ce^{-\alpha \theta _1(1-x)}e^{-\beta \theta _2(d-y)},\\ |p_{{8}}(x,y)| &{}\le Ce^{-\alpha \theta _2(x-d)}e^{-\beta \theta _2(d-y)},\\ \end{array}\right. }\\ {\left\{ \begin{array}{ll} |p_{{9}}(x,y)| &{}\le Ce^{-\alpha \theta _1x}e^{-\beta \theta _2(y-d)},\\ |p_{{10}}(x,y)| &{}\le Ce^{-\alpha \theta _2(d-x)}e^{-\beta \theta _2(y-d)},\\ |p_{{11}}(x,y)| &{}\le Ce^{-\alpha \theta _2(d-x)}e^{-\beta \theta _1(1-y)},\\ |p_{{12}}(x,y)| &{}\le Ce^{-\alpha \theta _1x}e^{-\beta \theta _1(1-y)},\\ \end{array}\right. }\\ {\left\{ \begin{array}{ll} |p_{{13}}(x,y)| &{}\le Ce^{-\alpha \theta _2(x-d)}e^{-\beta \theta _2(y-d)},\\ |p_{{14}}(x,y)| &{}\le Ce^{-\alpha \theta _1(1-x)}e^{-\beta \theta _2(y-d)},\\ |p_{{15}}(x,y)| &{}\le Ce^{-\alpha \theta _1(1-x)}e^{-\beta \theta _1(1-y)},\\ |p_{{16}}(x,y)| &{}\le Ce^{-\alpha \theta _2(x-d)}e^{-\beta \theta _1(1-y)},\\ \end{array}\right. } \end{aligned}$$
$$\begin{aligned} \text {where} \,\, \theta _1= {\left\{ \begin{array}{ll} \frac{\lambda }{\epsilon },\,\, \alpha \mu ^2\le \lambda \epsilon ,\\ \frac{ \mu ^2}{\epsilon ^2},\,\, \mu ^2>\lambda \epsilon ,\\ \end{array}\right. } \qquad \quad \theta _2= {\left\{ \begin{array}{ll} \frac{\lambda }{2\epsilon },\,\, \alpha \mu ^2\le \lambda \epsilon ,\\ \frac{ \lambda }{2\mu ^2},\,\, \alpha \mu ^2>\lambda \epsilon ,\\ \end{array}\right. } \end{aligned}$$
(2.10)
$$\begin{aligned} {\left\{ \begin{array}{ll} \bigg |\bigg |\dfrac{\partial ^i s_{{k}}}{\partial x^i}\bigg |\bigg | \le C(1+\epsilon ^{(1-i)}),\quad &{} \text {where} \,\,\, k =2,4,6,8,10,12,14,16 \,\, \, \text {and} \,\,\, 1\le i\le 4,\,\,\ \text {if} \,\, \alpha \mu ^2\le \lambda \epsilon ,\\ \bigg |\bigg |\dfrac{\partial ^i s_{{k}}}{\partial x^i}\bigg |\bigg | \le C\bigg (1+\bigg (\dfrac{\epsilon }{\mu }\bigg )^{2-2i}\bigg ),\quad &{} \text {where} \,\,\, k =2,6,12,16 \,\, \, \text {and} \,\,\, 1\le i\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon ,\\ \bigg |\bigg |\dfrac{\partial ^i s_{{k}}}{\partial x^i}\bigg |\bigg | \le C\bigg (1+\bigg (\dfrac{1}{\mu }\bigg )^{2i-2}\bigg ),\quad &{} \text {where} \,\,\, k = 4,8,10,14 \,\, \, \text {and} \,\,\, 1\le i\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon , \end{array}\right. } \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} \bigg |\bigg |\dfrac{\partial ^j s_{{k}}}{\partial y^j} \bigg |\bigg |\le C(1+\epsilon ^{(1-j)}), &{} \,\text {where} \,\,\,\, k =1,3,5,7,9,11,13,15 \,\, \, \text {and} \,\,\, 1\le j\le 4,\,\,\ \text {if} \,\, \alpha \mu ^2\le \lambda \epsilon ,\\ \bigg |\bigg |\dfrac{\partial ^j s_{{k}}}{\partial y^j} \bigg |\bigg |\le C\bigg (1+\bigg (\dfrac{\epsilon }{\mu }\bigg )^{2-2j}\bigg ), &{}\,\text {where} \,\,\,\,k =1,7,9,15 \,\, \, \text {and} \,\,\, 1\le j\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon ,\\ \bigg |\bigg |\dfrac{\partial ^j s_{{k}}}{\partial y^j} \bigg |\bigg |\le C\bigg (1+\bigg (\dfrac{1}{\mu }\bigg )^{2j-2}\bigg ), &{}\,\text {where} \,\,\,\,k =3,5,11,13 \,\, \, \text {and} \,\,\, 1\le j\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon , \end{array}\right. } \end{aligned}$$

\({\left\{ \begin{array}{ll} \max \bigg \{\bigg |\bigg | \dfrac{\partial ^{i+j} s_{{k}}}{\partial x^i \partial y^j}\bigg |\bigg |, \bigg |\bigg | \dfrac{\partial ^{i+j} p_{{k}}}{\partial x^i \partial y^j}\bigg |\bigg |\bigg \}\le C\epsilon ^{-(i+j)}, &{} 1\le i+j\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2\le \lambda \epsilon ,\\ \\ \bigg |\bigg | \dfrac{\partial ^{i+j} p_{{k}}}{\partial x^i \partial y^j}\bigg |\bigg |\le C\bigg (\dfrac{\epsilon }{\mu }\bigg )^{-(2i+2j)}, &{} 1\le i+j\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon ,\, k=1,6,12,15,\\ \\ \bigg |\bigg | \dfrac{\partial ^{i+j} p_{{k}}}{\partial x^i \partial y^j}\bigg |\bigg |\le C\bigg (\dfrac{1}{\mu }\bigg )^{2i}\bigg (\dfrac{\epsilon }{\mu }\bigg )^{-2j}, \quad &{} 1\le i+j\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon ,\, k=2,5,11,16,\\ \\ \bigg |\bigg | \dfrac{\partial ^{i+j} p_{{k}}}{\partial x^i \partial y^j}\bigg |\bigg |\le C\bigg (\dfrac{\epsilon }{\mu }\bigg )^{-2i}\bigg (\dfrac{1}{\mu }\bigg )^{2j}, &{} 1\le i+j\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon ,\, k=4,7,9,14,\\ \\ \bigg |\bigg | \dfrac{\partial ^{i+j} p_{{k}}}{\partial x^i\partial y^j}\bigg |\bigg |\le C\bigg (\dfrac{1}{\mu }\bigg )^{2i+2j}, &{} 1\le i\le 4, \,\,\ \text {if} \,\, \alpha \mu ^2> \lambda \epsilon ,\, k=3,8,10,13. \qquad \qquad \square \end{array}\right. }\)

The solution z(x, y) of (1.1) can be determined as follows

$$\begin{aligned} z(x,y)= {\left\{ \begin{array}{ll} (r_1 + s_{{1}} + s_{{2}} + s_{{3}} + s_{{4}} + p_{{1}} + p_{{2}} + p_{{3}} + p_{{4}})(x,y), &{} \forall (x,y) \in \Omega _1,\\ (r_2 + s_{{5}} + s_{{6}} + s_{{7}} + s_{{8}} + p_{{5}} + p_{{6}} + p_{{7}} + p_{{8}})(x,y), &{} \forall (x,y) \in \Omega _2,\\ (r_3 + s_{{9}} + s_{{10}} + s_{{11}} + s_{{12}} + p_{{9}} + p_{{10}} + p_{{11}} + p_{{12}})(x,y), &{} \forall (x,y) \in \Omega _3,\\ (r_4 + s_{{13}} + s_{{14}} + s_{{15}} + s_{{16}} + p_{{13}} + p_{{14}} + p_{{15}} + p_{{16}})(x,y), &{} \forall (x,y) \in \Omega _4,\\ {[}(r + s + p){]} (x,y)=0, \,\,[(r + s + p)_x](x,y)=0, &{} \forall (x,y) \in \Gamma _1,\\ {[}(r +s + p){]} (x,y) =0, \,\, [(r +s + p)_y] (x,y) =0, &{} \forall (x,y) \in \Gamma _2. \end{array}\right. } \end{aligned}$$

where, \(r = \sum\limits_{i=1}^{4}r_{i}\),  \(s = \sum\limits_{k=1}^{16}s_{{k}}\),  \(p = \sum\limits_{k=1}^{16}p_{{k}}\).

Discretization of the Problem

In this section, we introduce a piecewise uniform Shishkin mesh for problem (1.1) and use the finite-difference technique on this mesh to get the numerical solution. To construct a suitable fitted piecewise uniform mesh, we first subdivided the unit interval in both x and y directions into six subintervals as

$$\begin{aligned}{}[0,1]= [0,\sigma _1] \cup [\sigma _1,d-\sigma _2] \cup [d-\sigma _2,d] \cup [d,d+\sigma _2] \cup [d+\sigma _2,1-\sigma _1] \cup [1-\sigma _1,1]. \end{aligned}$$
(3.1)

On the subintervals \([\sigma _1,d-\sigma _2]\) and \([d+\sigma _2,1-\sigma _1]\) we take N/4 subdivisions, while in the remaining subintervals we take N/8 subdivisions on each (see, Fig. 3). The transition points, \(\sigma _1\) and \(\sigma _2,\) are defined as

Fig. 3
figure 3

Description of the Shishkin Mesh

Full size image
$$\begin{aligned} \sigma _1 = \min \bigg \{\frac{d}{4},\frac{2}{\theta _1}\ln N\bigg \},\qquad \sigma _2 = \min \bigg \{\frac{d}{4},\frac{2 }{\theta _2}\ln N\bigg \}, \end{aligned}$$
(3.2)

where \(\theta _1\) and \(\theta _2\) are defined in (2.10).

In view of the above, the corresponding step sizes on each direction are given by

$$\begin{aligned} h_1= & {} k_1= \dfrac{8\sigma _1}{N},\quad H_1=K_1= \dfrac{4(d-\sigma _1-\sigma _2)}{N}, \\ h_2= & {} k_2= \dfrac{8\sigma _2}{N},\quad H_2=K_2= \dfrac{4(1-\sigma _1-\sigma _2 - d)}{N}. \end{aligned}$$

The interior points of the mesh are denoted by \(\Omega ^{N,N}=\bigcup \limits _{k=1}^{4}\Omega _k^{N,N},\) where

$$\begin{aligned} \Omega _1{^{N,N}} =&\bigg \{(x_i,y_j):1\le i\le \frac{N}{2}-1, 1\le j\le \frac{N}{2}-1 \bigg \};\\ \Omega _2{^{N,N}} =&\bigg \{(x_i,y_j): \frac{N}{2}+1\le i\le N-1, 1\le j\le \frac{N}{2}-1 \bigg \};\\ \Omega _3^{{N,N}} =&\bigg \{(x_i,y_j): 1 \le i \le \frac{N}{2}-1, \frac{N}{2}+1\le j\le N-1 \bigg \};\\ \Omega _4^{{N,N}} =&\bigg \{(x_i,y_j) : \frac{N}{2}+1\le i\le N-1, \frac{N}{2}+1\le j\le N-1 \bigg \};\\ \Gamma _1^{N,N}=&\bigg \{(x_{N/2},y_j):1\le j \le N\bigg \};\quad \Gamma _2^{N,N}=\bigg \{(x_i,y_{N/2}):1\le i \le N\bigg \}. \end{aligned}$$

The boundaries of these subdomains are denoted as

$$\begin{aligned} \Lambda _{1}^{N,N}&=\bigg \{(0,y_j)\,\big |\,(0 \le j< N/2)\, \cup \, (N/2< j \le N) \bigg \},\\ \Lambda _{2}^{N,N}&=\bigg \{(x_i,0)\,\big |\,(0 \le i< N/2)\, \cup \, (N/2< i \le N) \bigg \},\\ \Lambda _{3}^{N,N}&=\bigg \{(1,y_j)\,\big |\,(0 \le j< N/2)\,\cup \, (N/2< j \le N)\bigg \},\\ \Lambda _{4}^{N,N}&=\bigg \{(x_i,1)\,\big |\,(0 \le i< N/2)\, \cup \, (N/2 < i \le N)\bigg \}, \end{aligned}$$

and \(\Lambda ^{N,N}=\Lambda _{1}^{N,N}\cup \Lambda _{2}^{N,N}\cup \Lambda _{3}^{N,N}\cup \Lambda _{4}^{N,N}.\) Note that \(\bar{\Omega }^{N,N}= \bigg \{(x_i,y_j):0\le i\le N, 0\le j\le N \bigg \}.\)

On an arbitrary mesh, \(\bar{\Omega }^{N,N}\), in order to discretized the problem (1.1) we define the standard upwind finite difference operator

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{\epsilon ,\mu }^{N,N}Z(x_i,y_j) \equiv \epsilon ^2(\delta _{xx}^2+\delta _{yy}^2)Z(x_i,y_j)+\mu ^2(a_{ij}D_x^*+b_{ij}D_y^*)Z(x_i,y_j)-c_{ij}Z(x_i,y_j)=f_{ij},\, \forall (x_i,y_j)\in {\Omega }^{N,N},\\ Z(x_i,y_j)=g_1(y_j),\,\,(x_i,y_j)\in \Lambda _1^{N,N},\quad Z(x_i,y_j)=g_2(x_i),\,\,(x_i,y_j)\in \Lambda _2^{N,N},\\ Z(x_i,y_j)=g_3(y_j),\,\,(x_i,y_j)\in \Lambda _3^{N,N},\quad Z(x_i,y_j)=g_4(x_i),\,\,(x_i,y_j)\in \Lambda _4^{N,N},\\ D_x^-Z(x_i,y_j)=D_x^+Z(x_i,y_j),\,\, (x_i,y_j)\in \Gamma _{1}^{N,N},\\ D_y^-Z(x_i,y_j)=D_y^+Z(x_i,y_j),\,\, (x_i,y_j)\in \Gamma _{2}^{N,N}. \end{array}\right. } \end{aligned}$$
(3.3)

Further, the discrete differential operators \(D_x^*,\,D_y^*,\, \delta _{xx}^2\), and \(\delta _{yy}^2\) are considered as follows:

$$\begin{aligned} D_x^*Z(x_i,y_j)&= {\left\{ \begin{array}{ll} D_x^+Z(x_i,y_j),\, i<N/2,\\ D_x^-Z(x_i,y_j),\, i>N/2, \end{array}\right. } &D_y^*Z(x_i,y_j)= {\left\{ \begin{array}{ll} D_y^+Z(x_i,y_j),\, j<N/2,\\ D_y^-Z(x_i,y_j),\, j>N/2, \end{array}\right. } \qquad\qquad\,\\[0.3cm] \delta ^2_{xx}Z(x_i,y_j)&= \frac{1}{\bar{h}_i}(D_x^+Z(x_i,y_j)-D_x^-Z(x_i,y_j)), &\delta ^2_{yy}Z(x_i,y_j)= \frac{1}{\bar{k}_j}(D_y^+Z(x_i,y_j)-D_y^-Z(x_i,y_j)), \quad\text { with}\\[0.3cm] D_x^+Z(x_i,y_j)&=\frac{Z(x_{i+1},y_j)-Z(x_{i},y_j)}{h_{i+1}},\\ D_x^-Z(x_i,y_j)&=\frac{Z(x_i,y_j)-Z(x_{i-1},y_j)}{h_i}. \end{aligned}$$

Lemma 3.1

(Discrete minimum principle): Let \(L_{\epsilon ,\mu }^{N,N}\) be the discrete operator given in (3.3), If \(\phi (x_i,y_j)\ge 0\) on \(\Lambda ^{N,N}\), \(L_{\epsilon ,\mu }^{N,N}\phi (x_i,y_j)\le 0\), \(\forall (x_i,y_j)\in \Omega ^{N,N}\), \((D_x^+\phi (x_i,y_j)-D_x^-\phi (x_i,y_j))\le 0,\, \forall (x_i,y_j)\in \Gamma _{1}^{N,N}\) and \((D_y^+\phi (x_i,y_j)-D_y^-\phi (x_i,y_j))\le 0,\, \forall (x_i,y_j)\in \Gamma _{2}^{N,N}\) then \(\phi (x_i,y_j)\ge 0\),   \(\forall (x_i,y_j)\in \bar{\Omega }^{N,N}.\)

Proof

We can prove the present Lemma using [15, 20]. \(\square\)

Lemma 3.2

(Discrete stability result) Let \(Z(x_i,y_j)\) be the solution of (3.3). Then it holds

$$\begin{aligned} ||Z(x_i,y_j)||_{\bar{\Omega }^{N,N}} \le \dfrac{1}{\nu } ||L_{\epsilon ,\mu }Z||_{\Omega ^{N,N}}+\max \bigg \{ || Z ||_{\Lambda _{1}^{N,N}},|| Z ||_{\Lambda _{2}^{N,N}},|| Z ||_{\Lambda _{3}^{N,N}},|| Z ||_{\Lambda _{4}^{N,N}}\bigg \}, \end{aligned}$$

where ||.|| denotes the pointwise maximum norm.

Proof

It can be proved using Lemma 3.1. \(\square\)

Error Analysis

   Lemma (3.2), will be used to proof the uniform convergence. Using standard techniques, the local truncation error may be readily bounded as

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N} (Z-z) (x_i,y_j) | \le {\left\{ \begin{array}{ll} C \epsilon ^2 \bigg (\bar{h}_i \bigg |\bigg | \dfrac{\partial ^3 z}{\partial x^3} \bigg |\bigg | + \bar{k}_j \bigg |\bigg | \dfrac{\partial ^3 z}{\partial y^3} \bigg |\bigg | \bigg )+C\mu ^2\bigg ({h}_{i+1} \bigg |\bigg | \dfrac{\partial ^2 z}{\partial x^2} \bigg |\bigg |&{}+{k}_{j+1} \bigg |\bigg | \dfrac{\partial ^2 z}{\partial y^2} \bigg |\bigg |\bigg ), \\ &{}\text {if } \, x_i = \sigma _1, d-\sigma _2, d+\sigma _2, 1-\sigma _1,\, \\&{} \text {or }\, y_j = \sigma _1,{} d-\sigma _2, d+\sigma _2, 1-\sigma _1, \\ C \epsilon ^2 \bigg ({h_i}^2 \bigg |\bigg | \dfrac{\partial ^4 z}{\partial x^4} \bigg |\bigg | + {k_j}^2 \bigg |\bigg | \dfrac{\partial ^4 z}{\partial y^4} \bigg |\bigg | \bigg )+C\mu ^2\bigg ({h}_i \bigg |\bigg | \dfrac{\partial ^2 z}{\partial x^2} \bigg |\bigg |&{}+{k}_{j} \bigg |\bigg | \dfrac{\partial ^2 z}{\partial y^2} \bigg |\bigg |\bigg ), \\ &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$
(4.1)

To get suitable bounds of this error, we decompose the discrete solution as

$$\begin{aligned} Z= \sum _{k=1}^{4} R_k + \sum _{l=1}^{16} S_{{l}} + \sum _{m=1}^{16} P_{{m}}, \end{aligned}$$

where \(R_k\) are the discrete regular components, \(S_{l}\) the discrete singular components, and \(P_{m}\) the discrete corner components. The three components are, respectively, solutions of the following problems

$$\begin{aligned} L^{N,N} _{\epsilon ,\mu }R_k(x_i,y_j) = f(x_i,y_j), \quad&\forall (x_i,y_j) \in \Omega ^{N,N}, \quad k=1,2,3,4,\end{aligned}$$
(4.2a)
$$\begin{aligned} R_k(x_i,y_j) = r_k(x_i,y_j), \quad&\forall (x_i,y_j) \in \Lambda ^{N,N},\end{aligned}$$
(4.2b)
$$\begin{aligned} \left[R_k\right](x_i,y_j)=0,\quad &\forall (x_i,y_j)\in \Gamma _1^{N,N}\cup \Gamma _2^{N,N},\end{aligned}$$
(4.2c)
$$\begin{aligned} \left[(R_k)_x\right](x_i,y_j)=0,\,\,\forall (x_i,y_j)\in \Gamma _{1}^{N,N},\, [(R_k)_y](x_i,y_j)=0,\,\,&\forall (x_i,y_j)\in \Gamma _{2}^{N,N}. \end{aligned}$$
(4.2d)
$$\begin{aligned} L^{N,N}_{\epsilon ,\mu } S_{l}(x_i,y_j) = 0, \quad&\forall (x_i,y_j) \in \Omega ^{N,N}, \quad l=1,...,16,\end{aligned}$$
(4.3a)
$$\begin{aligned} S_{l}(x_i,y_j) = s_{l}(x_i,y_j), \quad&\forall (x_i,y_j) \in \Lambda ^{N,N},\end{aligned}$$
(4.3b)
$$\begin{aligned} \left[S_l\right](x_i,y_j)=0,\quad&\forall (x_i,y_j)\in \Gamma _1^{N,N}\cup \Gamma _2^{N,N},\end{aligned}$$
(4.3c)
$$\begin{aligned} \left[(S_l)_x\right](x_i,y_j)=0,\,\,\forall (x_i,y_j)\in \Gamma _{1}^{N,N},\, [(S_l)_y](x_i,y_j)=0,\,\,&\forall (x_i,y_j)\in \Gamma _{2}^{N,N}. \end{aligned}$$
(4.3d)
$$\begin{aligned} L^{N,N}_{\epsilon ,\mu } P_{m}(x_i,y_j) = 0, \quad&\forall (x_i,y_j) \in \Omega ^{N,N}, \quad m=1,...,16,\end{aligned}$$
(4.4a)
$$\begin{aligned} P_{m}(x_i,y_j) = p_{m}(x_i,y_j), \quad&\forall (x_i,y_j) \in \Lambda ^{N,N},\end{aligned}$$
(4.4b)
$$\begin{aligned} \left[P_m\right](x_i,y_j)=0,\quad&\forall (x_i,y_j)\in \Gamma _1^{N,N}\cup \Gamma _2^{N,N},\end{aligned}$$
(4.4c)
$$\begin{aligned} \left[(P_m)_x\right](x_i,y_j)=0,\,\,\forall (x_i,y_j)\in \Gamma _{1}^{N,N},\, [(P_m)_y](x_i,y_j)=0,\,\,&\forall (x_i,y_j)\in \Gamma _{2}^{N,N}. \end{aligned}$$
(4.4d)

Using the result (2.5), and those from (2.3) and (4.2) we get the following straightforward estimate

\(| L^{N,N}_{\epsilon ,\mu }(R_k - r_k)(x_i,y_j) | \le\) \({\left\{ \begin{array}{ll} C N^{-1},\quad \text {if} \, x_i = \sigma _1, d-\sigma _2, d+\sigma _2, 1-\sigma _1, \, \text {or} \, y_j = \sigma _1, d-\sigma _2, d+\sigma _2, 1-\sigma _1,\\ C (N^{-2}+\epsilon N^{-1} ), \quad \text {otherwise}. \end{array}\right. }\)

Following [7, 19], we consider the barrier function

\(\Psi (x_i,y_j) = C N^{-2}(\Phi (x_i) + \Phi (y_j)) + C N^{-1}\),

where \(\Phi (\phi _i)\) is the piecewise-linear polynomial

$$\begin{aligned}&\Phi (\phi _i) = {\left\{ \begin{array}{ll} 1, \qquad \qquad &{}0 \le \phi _i \le \sigma _1,\\ 1-\frac{\phi _i-\sigma _1}{2(d-\sigma _1-\sigma _2)}, \qquad \,\, &{}\sigma _1 \le \phi _i \le d-\sigma _2,\\ \frac{d-\phi _i}{2\sigma _2}, \qquad \qquad &{}d-\sigma _2 \le \phi _i \le d,\\ \frac{\phi _i-d}{2\sigma _2}, \qquad \qquad &{}d \le \phi _i \le d+\sigma _2,\\ 1-\frac{\phi _i-d-\sigma _2}{2(1-d-\sigma _1-\sigma _2)}, \quad \,\, &{} d+\sigma _2 \le \phi _i \le 1-\sigma _1,\\ \frac{1-\phi _i}{2\sigma _1}, \quad \qquad &{}1-\sigma _1 \le \phi _i \le 1. \end{array}\right. } \end{aligned}$$

Noting that \(1/\sigma _2\ge 4,\) we have that

$$\begin{aligned}&{\epsilon ^2\delta ^2_x \Psi (x_i,y_j) = {\left\{ \begin{array}{ll} O(-N^{-1}\epsilon ^2), \quad &{} x_i = \sigma _1, d-\sigma _2, d+\sigma _2, 1-\sigma _1,\\ 0, \qquad &{} \text {otherwise}, \end{array}\right. }}\\[0.3cm]&{\epsilon ^2\delta ^2_y \Psi (x_i,y_j) = {\left\{ \begin{array}{ll} O(-N^{-1}\epsilon ^2), \quad &{} y_j = \sigma _1, d-\sigma _2, d+\sigma _2, 1-\sigma _1,\\ 0, \qquad &{}\text {otherwise}. \end{array}\right. }} \end{aligned}$$
$$\begin{aligned} D_x^{+}\Psi (x_i,y_j)\le 0,\,\, D_y^{+}\Psi (x_i,y_j)\le 0,\\ D_x^{-}\Psi (x_i,y_j)\ge 0,\,\, D_y^{-}\Psi (x_i,y_j)\ge 0. \end{aligned}$$

Combining this with Lemma 3.2 we get

$$\begin{aligned} || R_k - r_k || \le CN^{-1}, \quad k=1,2,3,4, \end{aligned}$$
(4.5)

which shows a suitable bound for the error of the regular components.

We utilize evidence-based on suitable barrier functions to show \(\epsilon\)-uniform bounds of the errors related to the corner and edge components. We consider the barrier functions as shown below: \({\left\{ \begin{array}{ll} G_{s_{1};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=1}^i(1+h_a\alpha {\theta _1})^{-1} \quad \quad , \quad i \ne 0, \quad 1 \le i< N/2, \\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=0, \end{array}\right. }\\ G_{s_{2};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=1}^j(1+k_a\beta {\theta _1})^{-1} \quad \quad , \quad j \ne 0, \quad 1 \le j< N/2,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=0, \end{array}\right. }\\ G_{s_{3};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=i+1}^{N/2}(1+h_a\alpha {\theta _2})^{-1} \quad , \quad i \ne N/2, \quad 0 \le i< N/2,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=N/2, \end{array}\right. }\\ G_{s_{4};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=j+1}^{N/2}(1+k_a\beta {\theta _2})^{-1} \quad , \quad j \ne N/2, \quad 0 \le j < N/2,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=N/2, \end{array}\right. } \end{array}\right. }\)

\({\left\{ \begin{array}{ll} G_{s_{5};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=N/2+1}^i(1+h_a\alpha {\theta _2})^{-1} \,\,, \quad i \ne N, \quad N/2+1 \le i< N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=N, \end{array}\right. }\\ G_{s_{6};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=1}^j(1+k_a\beta {\theta _1})^{-1}\quad \quad , \quad j \ne N/2, \quad 1 \le j< N/2,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=N/2, \end{array}\right. }\\ G_{s_{7};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=i+1}^{N}(1+h_a\alpha {\theta _1})^{-1} \quad , \quad i \ne N, \quad N/2+1 \le i< N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=N, \end{array}\right. }\\ G_{s_{8};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=j+1}^{N/2}(1+k_a\beta {\theta _2})^{-1} \quad , \quad j \ne N/2, \quad 0 \le j < N/2,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=N/2, \end{array}\right. } \end{array}\right. }\)

\({\left\{ \begin{array}{ll} G_{s_{9};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=1}^i(1+h_a\alpha {\theta _1})^{-1} \quad \quad , \quad i \ne 0, \quad 1 \le i< N/2,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=0, \end{array}\right. }\\ G_{s_{{10}};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=N/2+1}^j(1+k_a\beta {\theta _2})^{-1} \,, \quad j \ne N, \quad N/2+1 \le j< N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=N, \end{array}\right. }\\ G_{s_{{11}};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=i+1}^{N/2}(1+h_a\alpha {\theta _2})^{-1} \quad , \quad i \ne N/2, \quad 1 \le i< N/2,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=N/2, \end{array}\right. }\\ G_{s_{{12}};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=N/2+1}^{j}(1+k_a\beta {\theta _1})^{-1} \,, \quad j \ne N, \quad N/2+1 \le j < N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=N, \end{array}\right. } \end{array}\right. }\)

\({\left\{ \begin{array}{ll} G_{s_{{13}};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=N/2+1}^i(1+h_a\alpha {\theta _2})^{-1}, \quad i \ne N, \quad N/2+1 \le i< N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=N, \end{array}\right. }\\ G_{s_{{14}};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=N/2+1}^j(1+k_a\beta {\theta _2})^{-1}, \quad j \ne N, \quad N/2+1 \le j< N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=N, \end{array}\right. }\\ G_{s_{{15}};i} = {\left\{ \begin{array}{ll} \prod \limits _{a=i+1}^{N}(1+h_a\alpha {\theta _1})^{-1} \quad , \quad i \ne N, \quad N/2+1 \le i< N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad i=N, \end{array}\right. }\\ G_{s_{{16}};j} = {\left\{ \begin{array}{ll} \prod \limits _{a=N/2+1}^{j}(1+k_a\beta {\theta _1})^{-1} \,, \quad j \ne N, \quad N/2+1 \le j < N,\\ 1 \qquad \qquad \qquad \quad \,\, \quad , \quad j=N. \end{array}\right. } \end{array}\right. }\)

The above functions depict first-order Taylor estimates of the exponential functions associated with the singular components of problem (1.1). For all \(1\le i<N/2\), we have

\(\exp (-\alpha \theta _1 x_i) = \prod \limits _{a=1}^i \exp (-\alpha \theta _1 h_a) \le G_{s_{1,i}},\)

and for \(\sigma _1 < \frac{d}{8}\) and \(N/8 \le i \le N/2\) we have

$$\begin{aligned}&G_{s_{1,i}} \le G_{s_{1,N/8}} = \bigg ( 1 + \dfrac{16\, \ln N}{N}\bigg )^{-N/8} \le C N^{-1}, \end{aligned}$$
(4.6)
$$\begin{aligned}&L^{N}_{\epsilon ,\mu } G_{s_{1,i}} \le (\epsilon ^2 \alpha ^2\theta _1^2-\mu ^2 a(x_i,y_j)\alpha \theta _1-c(x_i,y_j)) G_{s_{1,i}}. \end{aligned}$$
(4.7)

Similar bounds can be obtained for the remaining edge functions.

Lemma 4.1

If \(s_{l}\) and \(S_{l}\) are the solutions of (2.8) and (4.3), respectively, then, for \(l=1,2,....,16,\)

$$\begin{aligned} |s_{l}(x_i,y_j) - S_{l}(x_i,y_j)| \le C(N^{-1} \ln N), \quad \text {if}\,\, \alpha \mu ^2\le \lambda \epsilon . \end{aligned}$$

Proof

If \(\sigma _1=d/4\) the proof can be obtained using standard techniques by taking into account that \(\epsilon ^{-2}\le C(\ln N)^2\). Thus, we will asume that \(\sigma _1<d/4\). Here we merely provide the specifications pertaining to the edge layer function \(s_{1}\). Similar results are valid for the remaining boundary layer functions. \(\square\)

From (4.3) and Theorem 2.9, it follows that

$$\begin{aligned} |S_{1}(x_i,y_j)|=|s_{1}(x_i,y_j)|\le C e^{-\alpha \theta _1 x_i}\le C G_{s_{1,i}},\quad (x_i,y_j)\in \Lambda ^{N,N}. \end{aligned}$$
(4.8)

Further, for all internal grid points \((x_i,y_j)\in \Omega _1^{N,N}\), from (4.3), (4.7), and the discrete minimum principle, we have

$$\begin{aligned} |S_{1}(x_i,y_j)|\le G_{s_{1,i}}. \end{aligned}$$
(4.9)

After applying Theorem 2.9 and (4.9), we conclude that

$$\begin{aligned} |s_{1}(x_i,y_j) - S_{1}(x_i,y_j)| \le |s_{1}(x_i,y_j)| + |S_{1}(x_i,y_j)| \le C G_{s_{1,i}}. \end{aligned}$$

Finally, from (4.6), we have

$$\begin{aligned} |s_{1}(x_i,y_j) - S_{1}(x_i,y_j)| \le C N^{-1}, \qquad N/8 \le i \le N/2, \,\, 0 \le j \le N/2. \end{aligned}$$
(4.10)

To get similar bounds of the error in the region \(\Omega ^{N,N}_{1,1} = \big \{(x_i,y_j) \,|\, 0< i< N/8, \, 0< j < N/2 \big \}\), we proceed as follows. Applying Taylor expansions, we get

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N} [S_{1} - s_{1}](x_i,y_j) | \le {\left\{ \begin{array}{ll} C \epsilon ^2\bigg ({h_i}^2 \bigg |\bigg | \dfrac{\partial ^4 s_{1}}{\partial x^4} \bigg |\bigg | + \bar{k}_j \bigg |\bigg | \dfrac{\partial ^3 s_{1}}{\partial y^3} \bigg |\bigg | \bigg )+C\mu ^2\bigg ({h}_i \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial x^2} \bigg |\bigg |+{k}_j \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial y^2} \bigg |\bigg |\bigg ), &{}\text {if} \, j = N/8,\, 3N/8,\\[0.2cm] C \epsilon ^2\bigg ({h_i}^2 \bigg |\bigg | \dfrac{\partial ^4 s_{1}}{\partial x^4} \bigg |\bigg | + {k_j}^2 \bigg |\bigg | \dfrac{\partial ^4 s_{1}}{\partial y^4} \bigg |\bigg | \bigg )+C\mu ^2\bigg ({h}_i \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial x^2} \bigg |\bigg |+{k}_j \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial y^2} \bigg |\bigg |\bigg ),&{} \text {otherwise}. \end{array}\right. } \end{aligned}$$
(4.11)

If \(\alpha \mu ^2\le \lambda \epsilon\), from Theorem 2.9, we have

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N}[S_{1}(x_i,y_j) - s_{1}(x_i,y_j)]| \le {\left\{ \begin{array}{ll} C N^{-1} \ln N+CN^{-1}, &{} \text {if}\,\, j = N/8,\, 3N/8,\\ CN^{-1} \ln N, &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

and using the discrete minimum principle and a suitable barrier function on \(\bar{\Omega }_{1,1}^{N,N},\) we obtain

$$\begin{aligned} |s_{1}(x_i,y_j) - S_{1}(x_i,y_j)| \le C N^{-1} \ln N, \quad (x_i,y_j) \in \bar{\Omega }_{1,1}^{N,N}. \end{aligned}$$
(4.12)

The result follows easily from (4.10) and (4.12).

Similar results can be obtained for the remaining boundary and interior layer functions \(s_{l},\, l=2,3,...,16.\)

Lemma 4.2

If \(s_{l}\) and \(S_{l}\) are the solutions of (2.8) and (4.3), respectively, then, for \(\alpha \mu ^2> \lambda \epsilon\) it holds

$$\begin{aligned}|s_{l}(x_i,y_j) - S_{l}(x_i,y_j)| \le {\left\{ \begin{array}{ll} CN^{-1} \ln ^2 N, \quad \text {if}\,\, k=1,2,6,7,9,12,15,16,\\ CN^{-1} \ln N, \quad \text {if}\,\,l= 3,4,5,8,10,11,13,14. \end{array}\right. } \end{aligned}$$

Proof

If \(\sigma _2=d/4\) the proof can be obtained using standard techniques by taking into account that \((\frac{1}{\mu })^2 \le C\ln N\) and \(\big (\frac{\mu }{\epsilon }\big )^2\le C\ln N\). Thus, we will asume that \(\sigma _2<d/4\). Here we merely provide the specifications pertaining to the edge layer function \(s_{1}\). Similar results can be obtained for the remaining boundary layer functions. \(\square\)

From (4.3) and Theorem 2.9, we have

$$\begin{aligned} |s_{1}(x_i,y_j)|=|s_{1}(x_i,y_j)|\le C e^{-\theta _1 x_i}\le C G_{s_{1,i}},\quad (x_i,y_j)\in \Lambda ^{N,N}. \end{aligned}$$
(4.13)

Further, for all internal grid points \((x_i,y_j)\in \Omega _1^{N,N}\), from (4.3), (4.7), and the discrete minimum principle, we have

$$\begin{aligned} |S_{1}(x_i,y_j)|\le G_{s_{1,i}}. \end{aligned}$$
(4.14)

Therefore, applying Theorem 2.9 and (4.14), we conclude that

$$\begin{aligned} |s_{1}(x_i,y_j) - S_{1}(x_i,y_j)| \le |s_{1}(x_,y_j)| + |S_{1}(x_,y_j)| \le C G_{s_{1,i}}. \end{aligned}$$

Therefore, from the corresponding result as in (4.6), we have

$$\begin{aligned} |s_{1}(x_i,y_j) - S_{1}(x_i,y_j)| \le C N^{-1}, \qquad N/8 \le i \le N/2, \,\, 0 \le j \le N/2. \end{aligned}$$
(4.15)

To get appropriate bounds for the error in the region \(\Omega ^{N}_{1,1}\), we proceed as follows. Applying Taylor series, we get

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N} [S_{1} - s_{1}](x_i,y_j) | \le {\left\{ \begin{array}{ll} C \epsilon ^2\bigg ({h_i}^2 \bigg |\bigg | \dfrac{\partial ^4 s_{1}}{\partial x^4} \bigg |\bigg | + \bar{k}_j \bigg |\bigg | \dfrac{\partial ^3 s_{1}}{\partial y^3} \bigg |\bigg | \bigg )+C\mu ^2\bigg ({h}_i \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial x^2} \bigg |\bigg | + \bar{k}_j \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial y^2} \bigg |\bigg |\bigg ), &{}\text {if} \, j = N/8,\, 3N/8,\\[0.2cm] C \epsilon ^2\bigg ({h_i}^2 \bigg |\bigg | \dfrac{\partial ^4 s_{1}}{\partial x^4} \bigg |\bigg | + {k_j}^2 \bigg |\bigg | \dfrac{\partial ^4 s_{1}}{\partial y^4} \bigg |\bigg | \bigg )+C\mu ^2\bigg ({h}_i \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial x^2} \bigg |\bigg | + {k}_j \bigg |\bigg | \dfrac{\partial ^2 s_{1}}{\partial y^2} \bigg |\bigg |\bigg ),&{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

If \(\alpha \mu ^2> \lambda \epsilon\), from Theorem 2.9, it follows that

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N}[S_{1}(x_i,y_j) - s_{1}(x_i,y_j)]| \le {\left\{ \begin{array}{ll} C \mu ^{4}\epsilon ^{-2}(N^{-1} \ln N+N^{-1}), &{} \text {if}\, j = N/8,\, 3N/8,\\ C \mu ^{4}\epsilon ^{-2}(N^{-1} \ln N), &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

After using the barrier function \(\mu ^2\epsilon ^{-2}(\sigma _1-x_i)\) to get a feasible bound on the error in the layer region \({\Omega }_{1,1}^{N,N},\) the application of the discrete minimum principle on \({\Omega }_{1,1}^{N,N},\) gives

$$\begin{aligned} |s_{1}(x_i,y_j) - S_{1}(x_i,y_j)| \le C N^{-1} \ln ^2 N, \quad (x_i,y_j) \in \bar{\Omega }_{1,1}^{N,N}. \end{aligned}$$
(4.16)

The result follows easily from (4.15) and (4.16).

Similarly, we can derive the corresponding bounds for the remaining boundary and interior layer functions \(s_{l},\, l=2,6,7,9,12,15,16.\)

If \(\alpha \mu ^2> \lambda \epsilon\), we examine the boundary layer function \(s_{3}.\) From (4.6-4.11) and Theorem 2.9, it follows that

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N}[S_{3}(x_i,y_j) - s_{3}(x_i,y_j)]| \le {\left\{ \begin{array}{ll} C (N^{-1} \ln N+N^{-1}), &{} \text {if}\, j = N/8,\, 3N/8,\\ C (N^{-1} \ln N), &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Therefore, the discrete minimum priciple, only on \(\bar{\Omega }_{2,1}^{N,N}=\{(x_i,y_j)\,|\, 3N/8\le i\le N/2,\, 0\le j\le N/2\},\) leads to

$$\begin{aligned} |s_{3}(x_i,y_j) - S_{3}(x_i,y_j)| \le C N^{-1} \ln N, \quad (x_i,y_j) \in \bar{\Omega }_{2,1}^{N,N}. \end{aligned}$$
(4.17)

We can proceed similarly to get appropriate bounds for the remaining boundary and interior layer functions \(s_l,\, l=4,5,8,10,11,13,14.\)

Lemma 4.3

If \(p_{m}\) and \(P_{m}\) are the solutions of (2.9) and (4.4), respectively, then for \(m=1,2,3,...,16,\) it holds

$$\begin{aligned} |p_{m}(x_i,y_j) - P_{m}(x_i,y_j)| \le {\left\{ \begin{array}{ll} C(N^{-1} \ln N), \quad \text {if}\,\, \alpha \mu ^2\le \lambda \epsilon ,\\ C(N^{-1} \ln ^2 N), \quad \text {if}\,\, \alpha \mu ^2> \lambda \epsilon . \end{array}\right. } \end{aligned}$$
(4.18)

Proof

Again, we merely provide the proof of (4.18) for the corner layer component \(p_{1}\) and in case of \(\sigma _1 < \frac{d}{4}\). Proceeding similarly as in Lemma 4.1, we get

$$\begin{aligned} |P_{1}(x_i,y_j)| \le C \min \{G_{s_{2,j}}, G_{s_{1,i}}\}, \quad \text {if} \,\,(x_i,y_j) \in \Lambda ^{N,N},\\ |p_{1}(x_i,y_j) - P_{1}(x_i,y_j)| \le C \min \{G_{s_{2,j}}, G_{s_{1,i}}\}, \quad \text {if} \,\,(x_i,y_j) \in \Omega ^{N,N}, \quad 0< i,j < N/2. \end{aligned}$$

Then, applying (4.6) we conclude that

$$\begin{aligned} |p_{1}(x_i,y_j) -P_{1}(x_i,y_j)| \le C N^{-1}, \qquad (x_i,y_j) \in \Omega ^{N,N} \backslash \Omega ^{N,N}_{1,2}, \end{aligned}$$
(4.19)

where, \(\Omega ^{N,N}_{1,2} = \{(x_i,y_j) \,|\,\, 0< i,j < N/8\}\). Ultimately, in \(\Omega ^{N,N}_{1,2}\) the truncation error satisfies

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N} [P_{1}(x_i,y_j) - p_{1}(x_i,y_j)]|&\le C \epsilon ^2\bigg ({h_i}^2 \bigg |\bigg | \dfrac{\partial ^4 p_{1}}{\partial x^4} \bigg |\bigg | + {{k}_j}^2 \bigg |\bigg | \dfrac{\partial ^4 p_{1}}{\partial y^4} \bigg |\bigg | \bigg )\\ {}&\quad +C\mu ^2\bigg ({h}_i \bigg |\bigg | \dfrac{\partial ^2 p_{1}}{\partial x^2} \bigg |\bigg | +{k}_j \bigg |\bigg | \dfrac{\partial ^2 p_{1}}{\partial y^2} \bigg |\bigg |\bigg ). \end{aligned}$$

If \(\alpha \mu ^2\le \lambda \epsilon\), from Theorem 2.9, it follows that

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N}[P_{1}(x_i,y_j) - p_{1}(x_i,y_j)]| \le C (N^{-1} \ln N), \end{aligned}$$

and using the discrete minimum principle on \(\bar{\Omega }_{1,2}^{N,N},\) we get

$$\begin{aligned} |p_{1}(x_i,y_j) - P_{1}(x_i,y_j)| \le C N^{-1} \ln N, \quad (x_i,y_j) \in \bar{\Omega }_{1,2}^{N,N}. \end{aligned}$$
(4.20)

The result follows from (4.19) and (4.20).

If \(\alpha \mu ^2>\lambda \epsilon\), from Theorem 2.9, it follows that

$$\begin{aligned} |L_{\epsilon ,\mu }^{N,N}[P_{1}(x_i,y_j) - p_{1}(x_i,y_j)]| \le C \mu ^{4}\epsilon ^{-2}(N^{-1} \ln N), \end{aligned}$$

Using the barrier function \(\mu ^2\epsilon ^{-2}(\sigma _1-x_i)\) to attain a feasible bound on the error in the layer region \({\Omega }_{1,2}^{N,N},\)

and the discrete minimum principle on \(\bar{\Omega }_{1,2}^{N,N}\), we obtain

$$\begin{aligned} |p_{1}(x_i,y_j) - P_{1}(x_i,y_j)| \le C N^{-1} \ln ^2 N, \quad (x_i,y_j) \in \bar{\Omega }_{1,2}^{N,N}. \end{aligned}$$
(4.21)

The result follows from (4.19) and (4.21).

The discrete solution \(Z(x_i,y_j)\) of (3.3) can be written as,

$$\begin{aligned} Z(x,y)= {\left\{ \begin{array}{ll} (R_1 + S_{{1}} + S_{{2}} + S_{{3}} + S_{{4}} + P_{{1}} + P_{{2}} + P_{{3}} + P_{{4}})(x,y), \qquad \qquad \, &{} \forall (x,y) \in {\Omega _1}^{N,N},\\ (R_2 + S_{{5}} + S_{{6}} + S_{{7}} + S_{{8}} + P_{{5}} + P_{{6}} + P_{{7}} + P_{{8}})(x,y), \qquad \qquad \,&{} \forall (x,y) \in {\Omega _2}^{N,N}, \\ (R_3 + S_{{9}} + S_{{10}} + S_{{11}} + S_{{12}} + P_{{9}} + P_{{10}} + P_{{11}} + P_{{12}})(x,y), \qquad \quad &{}\forall (x,y) \in {\Omega _3}^{N,N}, \\ (R_4 + S_{{13}} + S_{{14}} + S_{{15}} + S_{{16}} + P_{{13}} + P_{{14}} + P_{{15}} + P_{{16}})(x,y), \,\quad \quad &{} \forall (x,y) \in \Omega _4^{N,N}, \\ {[}(R + S + P){]} (x,y) = 0, \qquad \quad \quad \qquad \qquad \qquad \qquad &{}(x,y) \in \Gamma _1^{N,N},\\ {[}(R +S + P){]} (x,y) = 0, \qquad \qquad \qquad \quad \qquad \quad \qquad &{} (x,y) \in \Gamma _2^{N,N}. \end{array}\right. } \end{aligned}$$

where, \(R = \sum\limits_{k=1}^{4}R_{k}\),  \(S = \sum\limits_{l=1}^{16}S_{{l}}\),  \(P = \sum\limits_{m=1}^{16}P_{{m}}\).

Lemma 4.4

Let z be the solution of problem (1.1) and Z the numerical solution of (3.3) on the constructed piecewise-uniform Shishkin mesh. Then the error at the mesh points \((x_i,y_j) \in \bar{\Omega }^{N,N}\) satisfies

$$\begin{aligned}|Z(x_i,y_j)-z(x_i,y_j)| \le {\left\{ \begin{array}{ll} CN^{-1} \ln N, \quad \text {if}\,\, \alpha \mu ^2\le \lambda \epsilon ,\\ CN^{-1} \ln ^2 N, \quad \text {if}\,\, \alpha \mu ^2> \lambda \epsilon . \end{array}\right. } \end{aligned}$$

Proof

Combining Lemmas 4.1, 4.2 and 4.3, we obtain the following error bound for \(\Omega ^{N,N}\)

$$\begin{aligned}|Z(x_i,y_j)-z(x_i,y_j)| \le {\left\{ \begin{array}{ll} CN^{-1} \ln N, &{} \text {if}\,\, \alpha \mu ^2\le \lambda \epsilon ,\\ CN^{-1} \ln ^2 N, &{} \text {if}\,\, \alpha \mu ^2> \lambda \epsilon . \end{array}\right. } \end{aligned}$$

At the point \((x_i,y_j)=(d,y_j)\), we have \((D_x^+-D_x^-)Z(d,y_j).\) Therefore,

$$\begin{aligned} |(D_x^+-D_x^-)(Z-z)(d,y_j)|= & {} |(D_x^+-D_x^-)Z(d,y_j)-(D_x^+-D_x^-)z(d,y_j)|\\ {}\le & {} |(D_x^+-D_x^-)z(d,y_j)|. \end{aligned}$$

Now, note that \(h_2=\frac{8\sigma _2}{N}\) on either side of \((d,y_j).\) Therefore,

$$\begin{aligned} |(D_x^+-D_x^-)(Z-z)(d,y_j)|\le&|(D_x^+-D_x^-)z(d,y_j)|\\ \le&\bigg |(D_x^+-\frac{d}{dx})z(d^+,y_j)+(D_x^{-}-\frac{d}{dx})z(d^{-},y_j)\bigg |\\ \le&C\frac{h_2}{2}\bigg |\frac{\partial ^2z}{\partial x^2}\bigg |_{\Omega ^{N,N}}\\ |(D_x^+-D_x^-)(Z-z)(d,y_j)|\le&C{\left\{ \begin{array}{ll} \frac{h_2}{2}(\frac{1}{\epsilon })^2, \quad \text {if}\, \alpha \mu ^2\le \lambda \epsilon ,\\ \frac{h_2}{2}(\frac{1}{\mu })^2,\quad \text {if}\, \alpha \mu ^2> \lambda \epsilon . \end{array}\right. } \end{aligned}$$

At the point \((x_i,y_j)=(x_i,d)\), we have \((D_y^+-D_y^-)Z(x_i,d).\) Therefore,

$$\begin{aligned} |(D_y^+-D_y^-)(Z-z)(x_i,d)|= & {} |(D_y^+-D_y^-)Z(x_i,d)-(D_y^+-D_y^-)z(x_i,d)|\\ {}\le & {} |(D_y^+-D_y^-)z(x_i,d)|. \end{aligned}$$

Now, note that \(k_2=\frac{8\sigma _2}{N}\) on either side of \((x_i,d).\) Therefore,

$$\begin{aligned} |(D_y^+-D_y^-)(Z-z)(x_i,d)|\le&|(D_y^+-D_y^-)z(x_i,d)|\\ \le&\bigg |(D_y^+-\frac{d}{dy})z(x_i,d^+)+(D_y^{-}-\frac{d}{dy})z(x_i,d^{-})\bigg |\\ \le&C\frac{k_2}{2}\bigg |\frac{\partial ^2z}{\partial y^2}\bigg |_{\Omega ^{N,N}},\\ |(D_y^+-D_y^-)(Z-z)(x_i,d)|\le&C{\left\{ \begin{array}{ll} \frac{k_2}{2}(\frac{1}{\epsilon })^2, \quad \text {if}\, \alpha \mu ^2\le \lambda \epsilon ,\\ \frac{k_2}{2}(\frac{1}{\mu })^2,\quad \text {if}\, \alpha \mu ^2> \lambda \epsilon . \end{array}\right. } \end{aligned}$$

\(\square\)

Using the techniques given in [21, 22], we can also obtain the error for \((x_i,y_j)=(d,y_j)\cup (x_i,d)\). First, we obtain the result for the case \(\alpha \mu ^2\le \lambda \epsilon .\) Consider the following discrete barrier function:

$$\begin{aligned} \xi _1^{\pm }(x_i,y_j)= {\left\{ \begin{array}{ll} C_1(N^{-1}\ln N)+C_2\frac{h_2}{\epsilon }\sigma _2^2(x_i-(d-\sigma _2))\pm e(x_i,y_j),\,\, \text {for}\, x_i\in (d-\sigma _2,d),\\ C_3(N^{-1}\ln N)+C_4\frac{h_2}{\epsilon }\sigma _2^2((d+\sigma _2)-x_i)\pm e(x_i,y_j),\,\, \text {for}\, x_i\in (d,d+\sigma _2),\\ C_5(N^{-1}\ln N)+C_6\frac{k_2}{\epsilon }\sigma _2^2(y_j-(d-\sigma _2))\pm e(x_i,y_j),\,\, \text {for}\, y_j\in (d-\sigma _2,d),\\ C_7(N^{-1}\ln N)+C_8\frac{k_2}{\epsilon }\sigma _2^2((d+\sigma _2)-y_j)\pm e(x_i,y_j),\,\, \text {for}\, y_j\in (d,d+\sigma _2), \end{array}\right. } \end{aligned}$$

where \(h_2=\frac{8\sigma _2}{N}=k_2\). Then it is easy to verify that

$$\begin{aligned}\xi _1^\pm (x_i,y_j)\ge 0,\, \forall (x_i,y_j)\in \bar{\Omega }^{N,N}\cap \Lambda ^{N,N},\end{aligned}$$

for a suitable large \(C_1.\) We also have

$$\begin{aligned}L_{\epsilon ,\mu }^{N,N}\xi _1^\pm (x_i,y_j)\le 0,\,\forall (x_i,y_j)\in \Omega ^{N,N}\end{aligned}$$

and

$$\begin{aligned}(D_x^+-D_x^-)\xi _1^\pm (d,y_j)\le 0, \,\, (D_y^+-D_y^-)\xi _1^\pm (x_i,d)\le 0,\end{aligned}$$

for a suitable large \(C_2\). Thus, from the discrete comparison principle, we get

$$\begin{aligned}\xi _1^\pm (x_i,y_j)\ge 0,\,\forall (x_i,y_j)\in \bar{\Omega }^{N,N}.\end{aligned}$$

Therefore, for a suitable large N, we obtain the following estimate

$$\begin{aligned}|Z(x_i,y_j)-z(x_i,y_j)|\le C(N^{-1}\ln N),\, \text {if}\, \alpha \mu ^2\le \lambda \epsilon .\end{aligned}$$

In the second case, \(\alpha \mu ^2>\lambda \epsilon ,\) using a suitable barrier function and a similar procedure from the above techniques based on the discrete comparison principle, we also obtain the error estimate

$$\begin{aligned}|Z(x_i,y_j)-z(x_i,y_j)|\le C(N^{-1}\ln ^2 N),\, \text {if}\, \alpha \mu ^2> \lambda \epsilon .\end{aligned}$$

Hence, we have the required result.

Numerical Experiments

To illustrate the performance of the method developed in the preceding sections, we focused on two examples of the type in (1.1).

Example 5.1

\(\epsilon ^2 (z_{xx}(x,y) + z_{yy}(x,y))+\mu ^2 (a(x,y)z_{x}(x,y) + b(x,y)z_{y}(x,y)) - c(x,y) z(x,y) = f(x,y),\,\forall (x,y)\in \Omega ,\)

\(f_1(x,y)=-(0.5+xy/2);\,\, f_2(x,y)=(0.6+x+y); \,\,f_3(x,y)=(0.6+x+y);\,\, f_4(x,y)=-(0.5+xy/2);\,\, d=0.5,\)

where each \(f_k\) is defined over \(\Omega _k,\,k=1,2,3,4,\) with boundary conditions as well as convection and reaction coefficients

$$\begin{aligned} z(x,0)&= z(x,1) = z(0,y) = z(1,y) = 0,\\ a(x,y)&={\left\{ \begin{array}{ll} 1+\exp(xy),&{} x<d,\\ -1-\exp(xy),&{} x>d, \end{array}\right. } \quad b(x,y)= {\left\{ \begin{array}{ll} 1+\exp(xy),&{} y<d,\\ -1-\exp(xy),&{} y>d, \end{array}\right. } \quad c(x,y)=1+x^2+y^2. \end{aligned}$$

Example 5.2

\(\epsilon ^2 (z_{xx}(x,y) + z_{yy}(x,y))+\mu ^2 (a(x,y)z_{x}(x,y) +b(x,y) z_{y}(x,y)) - c(x,y) z(x,y) = f(x,y),\, \forall (x,y)\in \Omega\)

\(f_1(x,y)=-(1+2x+2y)=f_4(x,y);\,\, f_2(x,y)=-(4+3x+3y)=f_3(x,y);\,\, d=0.5,\)

where each \(f_k\) is defined over \(\Omega _k,\,k=1,2,3,4,\) with boundary conditions as well as convection and reaction coefficients

$$\begin{aligned} z (x,0)&= z(x,1) = z(0,y) = z(1,y) = 0,\\ a(x,y)&={\left\{ \begin{array}{ll} 2+x+y,&{} x<d,\\ -(3+2x-2y),&{} x>d, \end{array}\right. } \quad b(x,y)= {\left\{ \begin{array}{ll} 2+x+y,&{} y<d,\\ -(3+2x-2y),&{} y>d, \end{array}\right. } \quad c(x,y)=1+\exp(xy). \end{aligned}$$

The exact solutions of these problems are not known. Therefore, we use the double mesh principle explained in [23] to estimate the maximum point-wise error. It is given by

$$\begin{aligned} E^{N,N}_{\epsilon ,\mu } = \displaystyle \max _{(x_i,y_j)\in \bar{\Omega }^{N,N}} |{Z}^{2N,2N} (x_{2i},y_{2j})- Z^{N,N} (x_i,y_j)| \end{aligned}$$

where \({Z}^{2N,2N} (x_{2i},y_{2j})\) represents the numerical solution on a mesh with 2N subintervals on each direction. The parameter uniform maximum point-wise errors are determined applying the formula

$$\begin{aligned} E^{N,N} = \max _{\epsilon ,\mu } E^{N,N}_{\epsilon ,\mu }.\end{aligned}$$

The numerical order of convergence is given by

$$\begin{aligned} Q^{N,N} = log_2\bigg (\frac{E^{N,N}}{E^{2N,2N}}\bigg ).\end{aligned}$$

In Tables 1 and 2 we have choosen \(\epsilon =2^{-12}\) and \(\mu =2^{-(2k+1)},\,k=0,1,2,3...12\). These tables show the maximum point-wise errors and orders of convergence corresponding to Examples 5.1 and 5.2. Further, from this tables it is clear that our numerical scheme is almost first-order convergent as \(\mu \rightarrow 0\), which is the rate required (see, e.g., [24]) when dealing with reaction-diffusion problems. Figure 4 shows the numerical solution for \(\epsilon = 2^{-12},\, \mu =2^{-8}\), \(N=128\) and \(\epsilon = 2^{-12},\, \mu =2^{-4}\), \(N=128\) respectively corresponding to the test Example 5.1. Figure 5 shows the numerical solution for \(\epsilon = 2^{-8},\, \mu =2^{-5}\), \(N=128\) and \(\epsilon = 2^{-8},\, \mu =2^{-3}\), \(N=128\), respectively, corresponding to the test Example 5.2. Figures 6 and 7 show the Loglog plot for Examples 5.1 and 5.2, respectively, in which the second line from the below is represented the theoretical \(\epsilon\)-uniform error estimations, whereas the \(\epsilon\)-uniform error estimations \(E^{N,N}\) are represented by the first almost linear curve.

Fig. 4
figure 4

Surface graph of the numerical solution Z for Example 5.1

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Fig. 5
figure 5

Surface graph of the numerical solution Z for Example 5.2

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Fig. 6
figure 6

Loglog plot of the numerical solution Z for Example 5.1

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Fig. 7
figure 7

Loglog plot of the numerical solution Z for Example 5.2

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Table 1 Maximum point-wise errors \(E^{N,N}\) and orders of convergence \(Q^{N,N}\) for Example 5.1
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Table 2 Maximum point-wise errors \(E^{N,N}\) and orders of convergence \(Q^{N,N}\) for Example 5.2
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Conclusion

This study is concerned with two-parameter singularly perturbed steady-state 2-D convection-reaction-diffusion problems with interior layers. A finite-difference approach that yields almost first-order convergence is used to generate a parameter-uniform discrete solution. The analytical and discrete solutions are split into a sum of regular, singular, and corner components to address the convergence analysis and to obtain appropriate bounds. The numerical experiments show that the theoretical analysis agrees with the obtained results.