A parameter uniform numerical method for singularly perturbed delay problems with discontinuous convection coefficient

==Abstract==

In this paper a standard numerical method with piecewise linear interpolation on Shishkin mesh is suggested to solve singularly perturbed boundary value problem for second order ordinary delay differential equations with discontinuous convection coefficient and source term. An error estimate is derived by using the supremum norm and it is of almost first order convergence. Numerical results are provided to illustrate the theoretical results.

==Keywords==

Singularly perturbed problem; Convection–diffusion problem; Discontinuous convection coefficient; Shishkin mesh; Delay

==2010 Mathematics Subject Classification==

65L10; 65L11; 65L12

==1. Introduction==

Singularly perturbed ordinary differential equations with a delay are ordinary differential equations in which the highest derivative is multiplied by a small parameter and involving at least one delay term. Such type of equations arises frequently from the mathematical modelling of various practical phenomena, for example, in the modelling of the human pupil-light reflex  [[#br000070|[14]]], the study of bistable devices  [[#br000020|[4]]] and variational problems in control theory  [[#br000050|[10]]], etc. It is important to develop suitable numerical methods to solve singularly perturbed differential equations with a delay, whose accuracy does not depend on the parameter <math display="inline">\epsilon </math>, that is the methods are uniformly convergent with respect to the parameter.

In the past, only very few people had worked in the area Numerical Methods to Singularly Perturbed Delay Differential Equation(SPDDE). But in the recent years, there has been growing interest in this area. The authors of  [[#br000060|[12]]], [[#br000030|[6]]], [[#br000080|[16]]], [[#br000005|[1]]] and [[#br000010|[2]]] suggested some numerical methods for singularly perturbed delay differential equations with continuous data. Recently few authors in  [[#br000100|[20]]], [[#br000105|[21]]] and [[#br000085|[17]]] suggested some numerical method for singularly perturbed delay differential equations with discontinuous data.

In the present paper, as mentioned in the above abstract, motivated by the works of [[#br000035|[7]]], [[#br000015|[3]]] and [[#br000065|[13]]], we consider the following singularly perturbed boundary value problem [[#fd000010|(2.1)]] for second order ordinary delay differential equations with discontinuous convection coefficient and suggest a parameter uniform numerical method. It is proved that this method is uniformly convergent of order <math display="inline">O(N^{-1}ln^2N)</math>.

The present paper is organized as follows. In Section  [[#s000010|2]], the problem of study with discontinuous data is stated. Existence of the solution to the problem is established in Section  [[#s000015|3]]. A maximum principle of the DDE is established in Section  [[#s000020|4]]. Further a stability result is derived. Analytical results of the problem are derived in Section  [[#s000025|5]]. The present numerical method is described in Section  [[#s000030|6]] and an error estimate is derived in Section  [[#s000050|7]]. Section  [[#s000055|8]] presents numerical results.

<span id='s000010'></span>
==2. Statement of the problem==

Through out the paper, <math display="inline">C,C_1</math> denote generic positive constants independent of the singular perturbation parameter <math display="inline">\epsilon </math> and the discretization parameter <math display="inline">N</math> of the discrete problem. Further, <math display="inline">I_N</math> denotes <math display="inline">\lbrace 0,1,\ldots ,N\rbrace </math>. The supremum norm is used for studying the convergence of the numerical solution to the exact solution to a singular perturbation problem: <math display="inline">\Vert u\Vert _{\Omega }=sup_{x\in \Omega }\vert u(x)\vert </math>.

Motivated by the works of  [[#br000040|[8]]], [[#br000015|[3]]] and [[#br000065|[13]]], we consider the following BVP for SPDDE.

Find <math display="inline">u\in Y=C^0(\overline{\Omega })\cap C^1(\Omega )\cap C^2(\Omega ^{{_\ast}})</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
-\epsilon u^{{''}}(x)+a(x)u^{{'}}(x)+b(x)u(x-1)=f(x)\mbox{,}\quad x\in \Omega ^{{_\ast}}\mbox{,}\\
u(x)=\phi (x)\mbox{,}\quad x\in [-1,0]\mbox{,}\quad u(2)=l\mbox{,}
\end{array}</math>
|}
| style="text-align: right;" | (2.1)
|}

where <math display="inline">0<\epsilon \ll 1</math>, <math display="inline">a,f</math> are sufficiently smooth and bounded in <math display="inline">\Omega ^{{_\ast}}</math>. The function <math display="inline">b</math> is a sufficiently smooth function on <math display="inline">\overline{\Omega }</math>, <math display="inline">\Omega =(0,2)</math>, <math display="inline">\overline{\Omega }=[0,2]</math>, <math display="inline">\Omega ^{{_\ast}}=\Omega ^{-}\cup \Omega ^+</math>, <math display="inline">\Omega ^{-}=(0,1)</math>, <math display="inline">\Omega ^+=(1,2)</math> and <math display="inline">\phi </math> is smooth on <math display="inline">[-1,0]</math>.

The above problem [[#fd000010|(2.1)]] is equivalent to

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Pu(x):=\lbrace \begin{array}{cc}
-\epsilon u^{{''}}(x)+a_1(x)u^{{'}}(x)=f_1(x)-b(x)\phi (x-1)\mbox{,} & x\in \Omega ^{-}\mbox{,}\\
-\epsilon u^{{''}}(x)+a_2(x)u^{{'}}(x)+b(x)u(x-1)=f_2(x)\mbox{,} & x\in \Omega ^+\mbox{,}
\end{array}</math>
|}
| style="text-align: right;" | (2.2)
|}

where <math display="inline">u(1-)</math> and <math display="inline">u(1+)</math> denote the left and right limits of <math display="inline">u</math> at <math display="inline">x=1</math>, respectively.

<span id='s000015'></span>
==3. Existence result==

For the reader’s convenience some known results are briefly reported on this section and in Section  [[#s000020|4]]. They can be used here with some modifications.

==Theorem 3.1.                     ==

''The problem''    [[#fd000010|(2.1)]]   ''has a solution  ''<math display="inline">u\in C^0(\bar{\Omega })\cap C^1(\Omega )\cap C^2(\Omega ^{{_\ast}})</math>.

==Proof.                     ==

The proof is by construction. Let <math display="inline">y_1</math> and <math display="inline">y_2</math> be particular solutions of the DDEs,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>-\epsilon y_1^{{''}}(x)+a_1(x)y_1^{{'}}(x)+b(x)y_1(x-1)=f(x)\mbox{,}\quad x\in \Omega ^{-}\quad \mbox{and}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">y_1=\phi (x),x\in [-1,0]</math>, <math display="inline">a_1,a_2\in C^2(\bar{\Omega })</math> with the above properties.

Consider the function

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>y(x)=\lbrace \begin{array}{c}
y_1(x)+A\phi _1(x)\mbox{,}\quad x\in \Omega ^{-}\mbox{,}\\
y_2(x)+\phi _2(x)[u(2)-y_2(2)]+B\phi _3(x)\mbox{,}\quad x\in \Omega ^+\mbox{,}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">\phi _1</math>, <math display="inline">\phi _2</math> and <math display="inline">\phi _3</math> are the solutions of the following problems, respectively:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
-\epsilon \phi _1^{{''}}(x)+a_1(x)\phi _1^{{'}}(x)+b(x)\phi _1(x-1)=0\mbox{,}\quad x\in \Omega \mbox{,}\\
\phi _1(x)=0\mbox{,}\quad x\in [-1,0]\mbox{,}\quad \phi _1(2)=1\mbox{,}
\end{array}\lbrace \begin{array}{c}
-\epsilon \phi _2^{{''}}(x)+a_2(x)\phi _2^{{'}}(x)+b(x)\phi _2(x-1)=0\mbox{,}\quad x\in \Omega \mbox{,}\\
\phi _2(x)=0\mbox{,}\quad x\in [-1,0]\mbox{,}\quad \phi _2(2)=1\mbox{,}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
-\epsilon \phi _3^{{''}}(x)+a_2(x)\phi _3^{{'}}(x)+b(x)\phi _3(x-1)=0\mbox{,}\quad x\in \Omega \mbox{,}\\
\phi _3(x)=1\mbox{,}\quad x\in [-1,0]\mbox{,}\quad \phi _3(2)=0\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

It is easy to see that the above function <math display="inline">y</math> satisfies the differential equation  [[#fd000010|(2.1)]] and <math display="inline">u(0)=y(0)</math> and <math display="inline">u(2)=y(2)</math>. Using the similar arguments given in  [[#br000035|[7, Theorem 1]]], and [[#br000095|[19, Theorems 2,3]]] one can prove the existence of the solution. □

Note: For the existence of <math display="inline">\phi _i,i=1,2,3</math> one may refer to  [[#br000090|[18]]] and [[#br000025|[5]]].

<span id='s000020'></span>
==4. Stability result==

==Theorem 4.1 Maximum Principle.                     ==

''Let  ''<math display="inline">w\in C^0(\overline{\Omega })\cap C^2(\Omega ^{{_\ast}})</math>''be any function satisfying  ''<math display="inline">w(0)\geq 0</math>'',  ''<math display="inline">w(2)\geq 0</math>'',  ''<math display="inline">Pw(x)\geq 0,\forall x\in \Omega ^{{_\ast}}</math>''and  ''<math display="inline">w^{{'}}(1+)-w^{{'}}(1-)=[w^{{'}}](1)\leq 0</math>''. Then  ''<math display="inline">w(x)\geq 0,\forall x\in \overline{\Omega }</math>.

In the following we use the function

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>s(x)=\lbrace \begin{array}{c}
\frac{3}{2}+\frac{x}{2}\mbox{,}\quad x\in [0,1]\mbox{,}\\
3-x\mbox{,}\quad x\in [1,2]\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | (4.1)
|}

==Proof.                     ==

Using the above function <math display="inline">s</math> and the procedure adopted in  [[#br000100|[20, Theorem 3.1]]], one can prove this theorem. □

<span id='e000025'></span>

==Corollary 4.2 Stability Result.                     ==

''For any  ''<math display="inline">u\in Y</math>''we have''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\vert u(x)\vert \leq Cmax\lbrace \vert u(0)\vert ,\vert u(2)\vert ,\underset{\xi \in \Omega ^{{_\ast}}}{sup}\vert Pu(\xi )\vert \rbrace \mbox{,}\quad \forall x\in \overline{\Omega }\mbox{.}</math>
|}
| style="text-align: right;" | (4.2)
|}

==Proof.                     ==

Using the barrier function <math display="inline">\psi ^{\pm }(x)=CC_1s(x)\pm u(x),x\in \overline{\Omega }</math>, where <math display="inline">C_1=max\lbrace \vert u(0)\vert ,\vert u(2)\vert ,sup_{\xi \in \Omega ^{{_\ast}}}\vert Pu(\xi )\vert \rbrace </math> and the procedure adopted in  [[#br000100|[20, Theorem 3.2]]], we can prove this corollary. □

Note: An immediate consequence of the [[#e000025|Corollary 4.2]] is that, the solution of the BVP [[#fd000010|(2.1)]] is unique.

<span id='s000025'></span>
==5. Analytical results==

==Theorem 5.1.                     ==

''Let  ''<math display="inline">u</math>''be the solution of the problem''    [[#fd000010|(2.1)]]'', then we have the following bounds''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Vert u^{(k)}\Vert _{\Omega }\leq C\epsilon ^{-k}\mbox{,}\quad k=0,1\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

==Proof.                     ==

Let <math display="inline">x\in \Omega ^{-}</math>. Then we have,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle \int _0^xa(s)u^{'}(s)ds=[a(x)u(x)-a(0)u(0)]-\\\displaystyle -\int _0^xa^{'}(t)u(t)dt\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

Integrating [[#fd000040|(2.2)]] from 0 to <math display="inline">x</math> we get,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle -\epsilon (u^{'}(x)-u^{'}(0))=-\int _0^xa(t)u^{'}(t)dt+\int _0^x(f(t)-\\\displaystyle -b(t)\phi (t-1))dt=-[a(x)u(x)-a(0)u(0)]+\\\displaystyle +\int _0^x[a^{'}(t)u(t)+(f(t)-b(t)\phi (t-1))]dt\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

Therefore,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle \epsilon u^{'}(0)=\epsilon u^{'}(x)-[a(x)u(x)-a(0)u(0)]+\\\displaystyle +\int _0^x[a^{'}(t)u(t)+(f(t)-b(t)\phi (t-1))]dt\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

By the mean value theorem there exists a <math display="inline">z\in (0,\epsilon )</math> such that <math display="inline">\vert \epsilon u^{{'}}(z)\vert \leq 2\Vert u\Vert _{\overline{\Omega }}</math>. Therefore <math display="inline">\epsilon \vert u^{{'}}(0)\vert \leq C(\Vert u\Vert _{\overline{\Omega }}+\Vert f\Vert _{\Omega }+\Vert \phi \Vert _{[-1,0]})</math>. Hence,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\epsilon \vert u^{{'}}(x)\vert \leq Cmax\lbrace \Vert u\Vert _{\overline{\Omega }},\Vert f\Vert _{\Omega },\Vert \phi \Vert _{[-1,0]}\rbrace \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Similarly one can show that, <math display="inline">\epsilon \vert u^{{'}}(x)\vert \leq C,x\in \Omega ^+</math>.

From [[#fd000040|(2.2)]] it is easy to show that <math display="inline">\Vert u^{(k)}\Vert _{\Omega ^{{_\ast}}}\leq C\epsilon ^{-k},k=2,3</math>. Hence the proof. □

To derive uniform error estimates, we need sharper bounds on the derivatives of the solution <math display="inline">u</math>. We derive these using the following decomposition of the solution into smooth and singular components <math display="inline">u(x)=v(x)+w(x)</math> where <math display="inline">v</math> can be written in the form <math display="inline">v=v_0+\epsilon v_1+\epsilon ^2v_2</math> and <math display="inline">v_0,v_1</math> and <math display="inline">v_2</math> are defined respectively to be the solutions of the following problems:

Find <math display="inline">v_0\in C^0(\Omega ^{{_\ast}})\cap C^1(\Omega ^{{_\ast}})</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>a(x)v_0^{{'}}(x)+b(x)v_0(x-1)=f(x)\mbox{,}\quad x\in \Omega ^{{_\ast}}\mbox{,}</math>
|}
| style="text-align: right;" | (5.1)
|}

<math display="inline">v_1\in C^0(\Omega ^{{_\ast}})\cap C^1(\Omega ^{{_\ast}}\cup \lbrace 2\rbrace )</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>a(x)v_1^{{'}}(x)+b(x)v_1(x-1)=v_0^{{''}}(x)\mbox{,}\quad x\in \Omega ^{{_\ast}}\mbox{,}</math>
|}
| style="text-align: right;" | (5.3)
|}

and <math display="inline">v_2\in Y^{{_\ast}}</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Pv_2=v_1^{{''}}(x)\mbox{,}\quad x\in \Omega ^{{_\ast}}\mbox{,}</math>
|}
| style="text-align: right;" | (5.5)
|}

We assume that, <math display="inline">\Vert v_0^{{''}}\Vert _{\Omega ^{{_\ast}}}\leq C</math> and <math display="inline">\Vert v_1^{{'''}}\Vert _{\Omega ^{{_\ast}}}\leq C</math>.

Thus the smooth component <math display="inline">v</math> satisfies the following:

find <math display="inline">v\in C^0(\Omega ^{{_\ast}}\cup \lbrace 0,2\rbrace )\cap C^2(\Omega ^{{_\ast}})</math> such that

<span id='fd000170'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
Pv(x)=f(x)\mbox{,}\quad x\in \Omega ^{{_\ast}}\mbox{,}\\
v(x)=v_0(x)\mbox{,}\quad x\in [-1,0]\mbox{,}\quad v(2)=v_0(2)\mbox{,}\\
v(1)=v_0(1)+\epsilon v_1(1)+\epsilon ^2v_2(1)\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | (5.7)
|}

Further <math display="inline">w</math> satisfies the problem, that is, find <math display="inline">w\in C^0(\Omega ^{{_\ast}}\cup \lbrace 0,2\rbrace )\cap C^2(\Omega ^{{_\ast}})</math> such that

<span id='fd000175'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}Pw(x)=0\mbox{,}\quad x\in \Omega ^{_\ast }\mbox{,}\\ w(x)=0\mbox{,}\quad x\in [-1\\\displaystyle 0]\mbox{,}[w](1)=-[v](1)\mbox{,}[w^{'}](1)=-[v^{'}](1)\mbox{,}w(2)=0\mbox{.} \end{array}</math>
|}
| style="text-align: right;" | (5.8)
|}

Note that <math display="inline">v+w=u\in Y^{{_\ast}}</math>.

<span id='e000045'></span>

==Theorem 5.2.                     ==

''Let  ''<math display="inline">v</math>''and  ''<math display="inline">w</math>''be the solutions of the regular and singular components of the solution  ''<math display="inline">u</math>''. Then''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Vert v^{(k)}\Vert _{\Omega ^{{_\ast}}}\leq C(1+\epsilon ^{2-k})\mbox{,}\quad k=0,1,2,3,\vert w^{(k)}(x)\vert \leq C\lbrace \begin{array}{c}
\epsilon ^{-k}exp(-\alpha \frac{(1-x)}{\epsilon })\mbox{,}\quad x\in \Omega ^{-}\mbox{,}\\
\epsilon ^{-k}exp(-\alpha \frac{(x-1)}{\epsilon })+\epsilon ^{-k+1}exp(-\alpha \frac{(2-x)}{\epsilon })\mbox{,}\\
x\in \Omega ^+,k=0,1,2,3\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

==Proof.                     ==

Integrating the differential equation  [[#fd000130|(5.1)]]–[[#fd000150|(5.4)]] separately on <math display="inline">\Omega ^{-}</math> and <math display="inline">\Omega ^+</math>, we get <math display="inline">\Vert v_i\Vert \leq C,i=0,1</math> and by the stability result we have <math display="inline">\Vert v_2\Vert \leq C</math>. Therefore <math display="inline">\Vert v\Vert _{\Omega ^{{_\ast}}}\leq C</math>. Similarly one can prove that <math display="inline">\Vert v^{(k)}\Vert _{\Omega ^{{_\ast}}}\leq C(1+\epsilon ^{2-k}),k=0,1,2,3</math>.

Note that <math display="inline">\vert w(x)\vert \leq \vert u(x)\vert +\vert v(x)\vert </math>. From the stability result we have <math display="inline">\vert u(1)\vert \leq C</math>. Further, <math display="inline">\vert v(1)\vert \leq C</math>. Therefore <math display="inline">\vert w(1)\vert \leq \eta </math> (say). Now consider the barrier function

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\varphi _1^{\pm }(x)=\eta exp(\frac{-\alpha (1-x)}{\epsilon })\pm w(x)\mbox{,}\quad x\in [0,1]\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

It is easy to check that <math display="inline">\varphi _1^{\pm }(0)=\eta exp(\frac{-\alpha }{\epsilon })\pm w(0)\geq 0,\varphi _1^{\pm }(1)=\eta \pm w(1)\geq 0</math>.

Applying the result given in  [[#br000045|[9, Theorem 2.1]]] on <math display="inline">[0,1]</math>, we get <math display="inline">\varphi _1^{\pm }(x)\geq 0</math>.

Consider the barrier function

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle \varphi _2^{\pm }(x)=C_1(\epsilon +exp(-\alpha \frac{(x-1)}{\epsilon })-\epsilon exp(-\\\displaystyle -\alpha \frac{(2-x)}{\epsilon }))\pm w(x)\mbox{,}x\in [1,2]\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

It is easy to see that, <math display="inline">\varphi _2^{\pm }(1)=C_1(\epsilon +1-\epsilon exp(\frac{-\alpha }{\epsilon }))\pm w(1)\geq 0</math>, <math display="inline">\varphi _2^{\pm }(2)=C_1(\epsilon +exp(\frac{-\alpha }{\epsilon })-\epsilon )\pm w(2)\geq 0</math>. Again applying the result given in  [[#br000045|[9, Theorem 2.1]]] on <math display="inline">[1,2]</math>, then we get <math display="inline">\varphi _2^{\pm }(x)\geq 0</math>. Using the procedure adopted in  [[#br000035|[7, Lemma 4]]], one can prove the rest of this theorem. □

Note: From the above theorem it is easy to show that

<span id='fd000195'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\vert u(x)-v(x)\vert \leq C\lbrace \begin{array}{c}
exp(-\alpha (1-x)/\epsilon )\mbox{,}\quad x\in \Omega ^{-}\mbox{,}\\
exp(-\alpha (x-1)/\epsilon )+\epsilon exp(-\alpha (2-x)/\epsilon )\mbox{,}\quad x\in \Omega ^+\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | (5.9)
|}

<span id='s000030'></span>
==6. Discrete problem==

In this section, mesh selection strategy, namely piecewise uniform mesh (Shishkin mesh), is explained. Also upwind finite difference scheme with piecewise linear interpolation on Shishkin mesh for the problem [[#fd000010|(2.1)]] is described.

<span id='s000035'></span>
===6.1. Mesh selection strategy===

Since the BVP [[#fd000010|(2.1)]] exhibits strong interior layers at <math display="inline">x=1</math> and a weak boundary layer at <math display="inline">x=2</math>, we choose a piecewise uniform Shishkin mesh on <math display="inline">[0,2]</math>. For this we divide the interval <math display="inline">[0,2]</math> into five subintervals, namely <math display="inline">\Omega _1=[0,1-\tau _1]</math>, <math display="inline">\Omega _2=[1-\tau _1,1]</math>, <math display="inline">\Omega _3=[1,1+\tau _2]</math>,<math display="inline">\Omega _4=[1+\tau _2,2-\tau _2]</math>, <math display="inline">\Omega _5=[2-\tau _2,2]</math>, where <math display="inline">\tau _1=min\lbrace 0.5,\frac{2\epsilon lnN}{\alpha }\rbrace ,\tau _2=min\lbrace 0.25,\frac{2\epsilon lnN}{\alpha }\rbrace </math>. Let <math display="inline">h_1=4N^{-1}(1-\tau _1)</math>, <math display="inline">h_2=4N^{-1}\tau _1</math>, <math display="inline">h_3=8N^{-1}\tau _2</math>, <math display="inline">h_4=4N^{-1}(1-2\tau _2)</math>. The mesh <math display="inline">\overline{\Omega }^N=\lbrace x_0,x_1,\ldots ,x_N\rbrace </math> is defined by

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle x_0=0.0\mbox{,}\quad x_i=x_0+ih_1\mbox{,}\quad i=1(1)\frac{N}{4},\quad x_{i+\frac{N}{4}}=x_{\frac{N}{4}}+ih_2\mbox{,}\quad i=1(1)\frac{N}{4}\\\displaystyle x_{i+\frac{N}{2}}=x_{\frac{N}{2}}+ih_3\mbox{,}i=1(1)\frac{N}{8}\mbox{,}x_{i+\frac{5N}{8}}=x_{\frac{5N}{8}}+ih_4\mbox{,}i=1(1)\frac{N}{4}\\\displaystyle x_{i+\frac{7N}{8}}=x_{\frac{7N}{8}}+ih_3\mbox{,}i=1(1)\frac{N}{8}\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

===6.2. A finite difference scheme for [[#fd000040|(2.2)]]===

On <math display="inline">\overline{\Omega }^N</math>, we define the following scheme for the BVP [[#fd000040|(2.2)]]:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle P^NU(x_i)=-\epsilon \delta ^2U(x_i)+a(x_i)DU(x_i)+b(x_i)U^I(x_i)=\\\displaystyle =f^{_\ast }(x_i)\mbox{,}x_i\in \Omega ^{_\ast }\cap \overline{\Omega }^N\mbox{,}\end{array}</math>
|}
| style="text-align: right;" | (6.1)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\delta ^2U(x_i)=\frac{2[D^+U(x_i)-D^{-}U(x_i)]}{x_{i+1}-x_{i-1}}\mbox{,}\quad D^{-}U(x_i)=\frac{U(x_i)-U(x_{i-1})}{x_i-x_{i-1}},D^+U(x_i)=\frac{U(x_{i+1})-U(x_i)}{x_{i+1}-x_i}\mbox{,}\quad DU(x_i)=\lbrace \begin{array}{c}
D^{-}U(x_i)\mbox{,}\quad x_i\in \Omega ^{-}\cap \overline{\Omega }^N\mbox{,}\\
D^+U(x_i)\mbox{,}\quad x_i\in \Omega ^+\cap \overline{\Omega }^N\mbox{,}
\end{array}U^I(x_i)=\lbrace \begin{array}{c}
0\mbox{,}\quad x_i\in \Omega ^{-}\cap \overline{\Omega }^N\mbox{,}\\
U(x_j)\frac{x_{j+1}-(x_i-1)}{x_{j+1}-x_j}+U(x_{j+1})\frac{(x_i-1)-x_j}{x_{j+1}-x_j},x_i\in \Omega ^+\cap \overline{\Omega }^N,x_j\leq x_i-1\leq x_{j+1}\mbox{,}
\end{array}f^{{_\ast}}(x_i)=\lbrace \begin{array}{c}
f(x_i)-b(x_i)\phi (x_i-1)\mbox{,}\quad x_i\in \Omega ^{-}\cap \overline{\Omega }^N\mbox{,}\\
f(x_i)\mbox{,}\quad x_i\in \Omega ^+\cap \overline{\Omega }^N\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

===6.3. Discrete stability result===

<span id='e000055'></span>

==Lemma 6.1 Discrete Maximum Principle.                     ==

''Let  ''<math display="inline">Z(x_i)</math>''be a mesh function satisfying  ''<math display="inline">Z(x_0)\geq 0,Z(x_N)\geq 0</math>'',  ''<math display="inline">P^NZ(x_i)\geq 0,i\in I_N\setminus \lbrace 0,N/2,N\rbrace </math>''and  ''<math display="inline">(D^+-D^{-})Z(x_{N/2})=[DZ](x_{N/2})\leq 0</math>''. Then  ''<math display="inline">Z(x_i)\geq 0,\forall x_i\in \overline{\Omega }^N</math>.

==Proof.                     ==

Define <math display="inline">s(x_i)=\lbrace \begin{array}{c}
\frac{3}{2}+\frac{x_i}{2}\mbox{,}\quad x_i\in [0,1]\cap \overline{\Omega }^N\mbox{,}\\
3-x_i\mbox{,}\quad x_i\in [1,2]\cap \overline{\Omega }^N\mbox{.}
\end{array}</math>

Note that <math display="inline">P^Ns(x_i)>0,\forall x_i\in \Omega ^{{_\ast}}\cap \overline{\Omega }^N</math>, <math display="inline">[Ds](x_{N/2})<0</math>, <math display="inline">s(x_i)>0,\forall x_i\in \overline{\Omega }^N</math>.

Let <math display="inline">\mu ^{{_\ast}}=max\lbrace \frac{-Z(x_i)}{s(x_i)}:x_i\in \overline{\Omega }^N\rbrace </math>. Then there exists <math display="inline">x_i^{{_\ast}}\in \overline{\Omega }^N</math> such that <math display="inline">Z(x_i^{{_\ast}})+\mu ^{{_\ast}}s(x_i^{{_\ast}})=0</math> and <math display="inline">Z(x_i)+\mu ^{{_\ast}}s(x_i)\geq 0,\forall x_i\in \overline{\Omega }^N</math>. Therefore the mesh function <math display="inline">(Z+\mu ^{{_\ast}}s)</math> attains its minimum at <math display="inline">x_i=x_i^{{_\ast}}</math>. Suppose the theorem does not hold true, then <math display="inline">\mu ^{{_\ast}}>0</math>.

Case (i)<math display="inline">:(x_i^{{_\ast}}\in \Omega ^{-}\cap \overline{\Omega }^N)</math>

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle 0<P^N(Z+\mu s)(x_i^{_\ast })=-\epsilon \delta ^2(Z+\mu ^{_\ast }s)(x_i^{_\ast })+a_1(x_i^{_\ast })D^{-}(Z+\\\displaystyle +\mu ^{_\ast }s)(x_i^{_\ast })\leq 0\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

It is a contradiction.

Case (ii)<math display="inline">:(x_i^{{_\ast}}\in \Omega ^+\cap \overline{\Omega }^N)</math>

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle 0<P^N(Z+\mu s)(x_i^{_\ast })=-\epsilon \delta ^2(Z+\mu ^{_\ast }s)(x_i^{_\ast })+a_2(x_i^{_\ast })D^+\\\displaystyle +(Z+\mu ^{_\ast }s)(x_i^{_\ast })+b(x_i^{_\ast })(Z+\mu ^{_\ast }s)^I(x_i^{_\ast })\leq 0\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

It is a contradiction.

Case (iii)<math display="inline">:(x_i^{{_\ast}}=x_{N/2})</math>

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>0\leq [D(Z+\mu ^{{_\ast}}s)](x_{N/2})=[DZ](x_{N/2})+\mu ^{{_\ast}}[Ds](x_{N/2})<0\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

It is a contradiction. Hence the proof of the theorem. □

<span id='e000065'></span>

==Lemma 6.2.                     ==

''For any mesh function  ''<math display="inline">U(x_i)</math>''we have''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle \vert U(x_i)\vert \leq Cmax\lbrace \vert U(x_0)\vert ,\vert U(x_N)\vert \\\displaystyle \underset{j\in I_N\setminus \lbrace 0,N/2,N\rbrace }{max}P^NU(x_j)\rbrace \mbox{,}x_i\in \overline{\Omega }^N\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

==Proof.                     ==

One can easily prove this lemma by using [[#e000055|Lemma 6.1]] and the discrete barrier function <math display="inline">\varphi ^{\pm }(x_i)=CC_1s(x_i)\pm U(x_i),x_i\in \overline{\Omega }^N</math>, where <math display="inline">\begin{array}{l}\displaystyle C_1=max\lbrace \vert U(x_0)\vert ,\vert U(x_N)\vert \\\displaystyle max_{j\in I_N\setminus \lbrace 0,N/2,N\rbrace }P^NU(x_j)\rbrace \end{array}</math>. □

Analogous to the continuous function <math display="inline">u</math>, we decompose the numerical solution <math display="inline">U(x_i)</math> defined by [[#fd000210|(6.1)]]–[[#fd000220|(6.3)]] as <math display="inline">U(x_i)=V(x_i)+W(x_i)</math>, where <math display="inline">V(x_i)</math> and <math display="inline">W(x_i)</math> satisfy the following:

<span id='fd000250'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
P^NV(x_i)=f^{{_\ast}}(x_i)\mbox{,}\quad i\in I_N\setminus \lbrace 0,N/2,N\rbrace \mbox{,}\\
V(x_0)=v(0)\mbox{,}\quad [D]V(x_{N/2})=[v^{{'}}](1)\mbox{,}\quad V(x_N)=v(2)\mbox{,}
\end{array}</math>
|}
| style="text-align: right;" | (6.4)
|}

and

<span id='fd000255'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
P^NW(x_i)=0\mbox{,}\quad i\in I_N\setminus \lbrace 0,N/2,N\rbrace \mbox{,}\\
W(x_0)=w(0)\mbox{,}\quad W(x_N)=w(2)\mbox{,}\quad [D]W(x_{N/2})=-[D]V(x_{N/2})\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | (6.5)
|}

<span id='e000075'></span>

==Theorem 6.3.                     ==

''Let  ''<math display="inline">U(x_i)</math>''be the numerical solution of''    [[#fd000040|(2.2)]]   ''defined by''    [[#fd000210|(6.1)]]''–''[[#fd000220|(6.3)]]   ''and further let  ''<math display="inline">V(x_i)</math>''be the numerical solution of''    [[#fd000170|(5.7)]]   ''given by''    [[#fd000250|(6.4)]]''. Then,''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\vert U(x_i)-V(x_i)\vert \leq C\lbrace \begin{array}{c}
N^{-1}\mbox{,}\quad i\in I_N\setminus \lbrace N/4+1,\ldots ,5N/8-1\rbrace \\
N^{-1}+\vert U(x_{N/2})-V(x_{N/2})\vert \mbox{,}\quad \mbox{otherwise  .}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

==Proof.                     ==

Consider a mesh function <math display="inline">\begin{array}{l}\displaystyle \varphi ^{\pm }(x_i)=C_1N^{-1}[s(x_i)+\eta (x_i)]+C_1\psi (x_i)\vert U(x_{N/2})-\\\displaystyle -V(x_{N/2})\vert \pm (U(x_i)-V(x_i)),i\in I_N\end{array}</math> where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>s(x_i)=\lbrace \begin{array}{cc}
\frac{3}{2}+\frac{x_i}{2}\mbox{,} & x_i\in \Omega ^{-}\cap \bar{\Omega }^N\mbox{,}\\
3-x_i\mbox{,} & x_i\in \Omega ^+\cap \bar{\Omega }^N\mbox{,}
\end{array}\mbox{,}\quad \eta (x_i)=\lbrace \begin{array}{cc}
2+x_i\mbox{,} & x_i\in \Omega ^{-}\cap \bar{\Omega }^N\mbox{,}\\
2-x_i\mbox{,} & x_i\in \Omega ^+\cap \bar{\Omega }^N\mbox{,}
\end{array}\psi (x_i)=\lbrace \begin{array}{cc}
0\mbox{,} & i\in I_N\setminus \lbrace N/4+1,\ldots ,5N/8-1\rbrace \\
\eta (x_i)\vert U(x_{N/2})-V(x_{N/2})\vert \mbox{,} & i=N/4+1,\ldots ,5N/8-1\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

It is easy to see that, <math display="inline">\varphi ^{\pm }(x_0)\geq 0</math> and <math display="inline">\varphi ^{\pm }(x_N)\geq 0</math> for a suitable <math display="inline">C_1>0</math>. Further,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle P^N\varphi ^{\pm }(x_i)=C_1[a(x_i)N^{-1}(1+\frac{s(x_i)-s(x_{i-1})}{x_i-x_i-1})]+\\\displaystyle +C_1[a(x_i)D^{-}\psi (x_i)]\pm P^N(U(x_i)-V(x_i))\mbox{,}x_i\in \Omega ^{-}\cap \bar{\Omega }^N\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | 
|}

Note that, for <math display="inline">i\in I_N\setminus \lbrace 0,N/2,N\rbrace </math>, we have <math display="inline">P^N(U(x_i)-V(x_i))=0</math>.

Hence <math display="inline">P^N\varphi ^{\pm }(x_i)\geq 0,i\in I_N\setminus \lbrace 0,N/2,N\rbrace </math> by a proper choice of <math display="inline">C_1</math>.

Let <math display="inline">x_i=x_{N/2}</math>, then <math display="inline">\begin{array}{l}\displaystyle [D]\varphi ^{\pm }(x_i)=-C_1\frac{7N^{-1}}{2}-C_12\vert U(x_{N/2})-\\\displaystyle -V(x_{N/2})\vert \pm [[D]U(x_i)-[D]V(x_i)]\leq 0\end{array}</math>, by [[#fd000215|(6.2)]], [[#fd000250|(6.4)]] and <math display="inline">\begin{array}{l}\displaystyle (x_{i+1}-\\\displaystyle -x_{i-1})\vert \delta ^2v(x_i)\vert \leq max_{[x_{i-1},x_{i+1}]}\vert v^{''}(x)\vert N^{-1}\end{array}</math>[[#br000075|[15, page 52]]]. Then by [[#e000055|Lemma 6.1]], we have <math display="inline">\varphi ^{\pm }(x_i)\geq 0,\forall i\in I_N</math>. Hence the proof. □

<span id='s000050'></span>
==7. Error analysis==

In this section we derive an error estimate for the numerical solution obtained by the scheme [[#fd000210|(6.1)]]–[[#fd000220|(6.3)]] for the problem [[#fd000010|(2.1)]].

<span id='e000085'></span>

==Lemma 7.1.                     ==

''Let  ''<math display="inline">v</math>''be the solution of the problem''    [[#fd000170|(5.7)]]   ''and let  ''<math display="inline">V(x_i)</math>''be its numerical solution defined by''    [[#fd000250|(6.4)]]''. Then,  ''<math display="inline">\vert v(x_i)-V(x_i)\vert \leq CN^{-1},i\in I_N</math>.

==Proof.                     ==

Now,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>P^N(v(x_i)-V(x_i))=-\epsilon (\delta ^2-\frac{d^2}{dx^2})v(x_i)+a(x_i)(D^{-}-\frac{d}{dx})v(x_i)+b(x_i)\lbrace \begin{array}{c}
0\mbox{,}\quad i=1,2,\ldots ,N/2-1\mbox{,}\\
v^I(x_i)-v(x_i-1)\mbox{,}\quad i=N/2+1,\ldots ,N-1\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

Since <math display="inline">\vert v^I(x_i)-v(x_i-1)\vert \leq CN^{-2}</math>   [[#br000055|[11]]], then <math display="inline">\vert P^N(v(x_i)-V(x_i))\vert \leq CN^{-1},i\in I_N\setminus \lbrace 0,N/2,N\rbrace </math>. Then by [[#e000065|Lemma 6.2]], we have <math display="inline">\vert v(x_i)-V(x_i)\vert \leq CN^{-1},i\in I_N</math>. Hence the proof. □

<span id='e000095'></span>

==Lemma 7.2.                     ==

''Let  ''<math display="inline">w</math>''be the solution to the problem''    [[#fd000175|(5.8)]]   ''and let  ''<math display="inline">W(x_i)</math>''be its numerical solution defined by''    [[#fd000255|(6.5)]]''. If  ''<math display="inline">\epsilon \leq CN^{-1}</math>'', then we have  ''<math display="inline">\vert w(x_i)-W(x_i)\vert \leq CN^{-1}ln^2N</math>'',  ''<math display="inline">i\in I_N</math>.

==Proof.                     ==

Note that <math display="inline">\vert w(x_i)-W(x_i)\vert \leq \vert u(x_i)-U(x_i)\vert +\vert v(x_i)-V(x_i)\vert </math>. Then by Eq. [[#fd000195|(5.9)]], [[#e000075|Theorem 6.3]] and [[#e000085|Lemma 7.1]], we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\vert u(x_i)-U(x_i)\vert \leq \vert U(x_i)-V(x_i)\vert +\vert v(x_i)-V(x_i)\vert +\vert u(x_i)-v(x_i)\vert \leq C\lbrace \begin{array}{c}
N^{-1}+exp(-\alpha \tau _1/\epsilon )\mbox{,}\quad i=0,1,\ldots ,N/4\mbox{,}\\
N^{-1}+\vert U(x_{N/2})-V(x_{N/2})\vert +exp(-\alpha (1-x_i)/\epsilon )\mbox{,}\\
i=N/4+1,\ldots ,N/2\mbox{,}\\
N^{-1}+\vert U(x_{N/2})-V(x_{N/2})\vert +exp(-\alpha (x_i-1)/\epsilon )\mbox{,}\\
i=N/2+1,\ldots ,5N/8\mbox{,}\\
N^{-1}+exp(-\alpha \tau _2/\epsilon )\mbox{,}\quad i=5N/8+1,\ldots ,N\mbox{,}
\end{array}\leq C\lbrace \begin{array}{c}
N^{-1}\mbox{,}\quad i=0,1,\ldots ,N/4\mbox{,}\\
N^{-1}+\vert U(x_{N/2})-V(x_{N/2})\vert +exp(-\alpha (1-x_i)/\epsilon )\mbox{,}\\
i=N/4+1,\ldots ,N/2\mbox{,}\\
N^{-1}+\vert U(x_{N/2})-V(x_{N/2})\vert +exp(-\alpha (x_i-1)/\epsilon )\mbox{,}\\
i=N/2+1,\ldots ,5N/8\mbox{,}\\
N^{-1}\mbox{,}\quad i=5N/8+1,\ldots ,N\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

Therefore

<span id='fd000295'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle \vert w(x_i)-W(x_i)\vert \leq \vert u(x_i)-U(x_i)\vert +\vert v(x_i)-\\\displaystyle -V(x_i)\vert \leq CN^{-1}\mbox{,}i=0,1,\ldots,N/4,5N/8,\ldots,N\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | (7.1)
|}

Now consider a mesh function

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\varphi ^{\pm }(x_i)=\lbrace \begin{array}{c}
C_1N^{-1}[[2+x_i]+\frac{\tau }{\epsilon ^2}[x_i-1+\tau _1]]\pm (w(x_i)-W(x_i))\mbox{,}\\
x_i\in [1-\tau _1,1)\cap \overline{\Omega }^N\mbox{,}\\
C_1N^{-1}[[2-x_i]+\frac{\tau }{\epsilon ^2}[1+\tau _2-x_i]]\pm (w(x_i)-W(x_i))\mbox{,}\\
x_i\in [1,1+\tau _2]\cap \overline{\Omega }^N\mbox{,}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">\tau =min\lbrace \tau _1,\tau _2\rbrace </math>. From [[#fd000295|(7.1)]], it is clear that <math display="inline">\varphi ^{\pm }(x_{N/4})\geq 0</math> and <math display="inline">\varphi ^{\pm }(x_{5N/8})\geq 0</math> for a suitable choice of <math display="inline">C_1>0</math>.

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>P^N\varphi ^{\pm }(x_i)=\lbrace \begin{array}{c}C_1N^{-1}a_1[1+\frac{\tau }{\epsilon ^2}]\pm P^N(w(x_i)-W(x_i))\mbox{,}\quad x_i\in [1-\tau _1,1)\cap \overline{\Omega }^N\mbox{,}\\ C_1N^{-1}[a_2[-1-\frac{\tau }{\epsilon ^2}]+b(x_i)(2-x_i)^I+\frac{\tau }{\epsilon ^2}[1+\tau _2-\\\displaystyle -x_i]^I]\pm P^N(w(x_i)-W(x_i))\mbox{,}x_i\in (1,1+\tau _2]\cap \overline{\Omega }^N \end{array}\geq \lbrace \begin{array}{c}C_1N^{-1}\alpha [1+\frac{\tau }{\epsilon ^2}]\pm P^N(w(x_i)-W(x_i))\mbox{,}\quad x_i\in [1-\tau _1,1)\cap \overline{\Omega }^N\mbox{,}\\ C_1N^{-1}[\alpha +2\beta _0][1+\frac{\tau }{\epsilon ^2}]\pm P^N(w(x_i)-W(x_i))\mbox{,}\\ x_i\in (1,1+\tau _2]\cap \overline{\Omega }^N\mbox{.} \end{array}</math>
|}
| style="text-align: right;" | 
|}

Note that,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>P^N(w(x_i)-W(x_i))=P^Nw(x_i)-P^NW(x_i)=\lbrace \begin{array}{c}
-\epsilon (\delta ^2-\frac{d^2}{dx^2})w(x_i)+a_1(x_i)(D^{-}-\frac{d}{dx})w(x_i)\mbox{,}\\
x_i\in [1-\tau _1,1)\cap \overline{\Omega }^N\mbox{,}\\
-\epsilon (\delta ^2-\frac{d^2}{dx^2})w(x_i)+a_2(x_i)(D^+-\frac{d}{dx})w(x_i)\\
\quad +b(x_i)[w^I(x_i)-w(x_i-1)]\mbox{,}\quad x_i\in (1,1+\tau _2]\cap \overline{\Omega }^N\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | 
|}

Also note that, <math display="inline">\vert w^I(x_i)-w(x_i-1)\vert \leq CN^{-1}</math>   [[#br000055|[11]]]. Further, <math display="inline">\vert P^N(w(x_i)-W(x_i))\vert \leq C_2\epsilon ^{-2}N^{-1}</math>, where <math display="inline">C_2>0</math> a constant independent of <math display="inline">\epsilon </math> and <math display="inline">N</math>.

Therefore, <math display="inline">P^N\varphi ^{\pm }(x_i)\geq 0,i\in \lbrace N/4+1,\ldots ,N/2-1,N/2+1,\ldots ,5N/8-1\rbrace </math>. Then by the [[#e000055|Lemma 6.1]], we have <math display="inline">\vert w(x_i)-W(x_i)\vert \leq CN^{-1}ln^2N,i=N/4+1,\ldots ,5N/8-1</math>. Hence the proof. □

==Theorem 7.3.                     ==

''Let  ''<math display="inline">u</math>''be the solution of the problem''    [[#fd000040|(2.2)]]'',  ''<math display="inline">U(x_i)</math>''be its numerical solution defined by''    [[#fd000210|(6.1)]]''–''[[#fd000220|(6.3)]]''. Then  ''<math display="inline">\vert u(x_i)-U(x_i)\vert \leq CN^{-1}ln^2N,i\in I_N</math>.

==Proof.                     ==

The desired estimate follows from the fact that <math display="inline">u=v+w,U=V+W</math> and from the [[#e000085|Lemma 7.1]] and [[#e000095|Lemma 7.2]]. □

<span id='s000055'></span>
==8. Numerical examples==

In this section, three examples are given to illustrate the numerical method discussed in this paper. We use the double mesh principle to estimate the error and compute the experiment rate of convergence in our computed solutions for all problems. For this we put <math display="inline">D_{\epsilon }^M=max_{0\leq i\leq M}\vert U_i^M-U_{2i}^{2M}\vert </math>, where <math display="inline">U_i^M</math> and <math display="inline">U_{2i}^{2M}</math> are the <math display="inline">i</math>th components of the numerical solutions on meshes of <math display="inline">M</math> and <math display="inline">2M</math> points respectively. We compute the uniform error and rate of convergence as <math display="inline">D^M=max_{\epsilon }D_{\epsilon }^M\mbox{and}\quad p^M=log_2(\frac{D^M}{D^{2M}})</math>. For the following examples the numerical results are presented for the values of perturbation parameter <math display="inline">\epsilon \in \lbrace 2^{-27},2^{-12},\cdots ,2^{-6}\rbrace </math>.

<span id='e000115'></span>

==Example 8.1.                     ==

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
-\epsilon u^{{''}}(x)+3u^{{'}}(x)-u(x-1)=0\mbox{,}\quad x\in \Omega ^{-}\\
-\epsilon u^{{''}}(x)-4u^{{'}}(x)-u(x-1)=0\mbox{,}\quad x\in \Omega ^+\\
u(x)=1\mbox{,}\quad x\in [-1,0]\mbox{,}\quad u(2)=2\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | (8.1)
|}

[[#t000005|Table 1]] presents the values of <math display="inline">D^N</math> and <math display="inline">p^N</math> for this problem. [[#f000005|Fig. 1]] and [[#f000010|Fig. 2]] represent the numerical solution and the maximum point wise error for this problem, respectively.

<span id='t000005'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 1.

Numerical results for the problem stated in [[#e000115|Example 8.1]].

|-

! <math display="inline">\epsilon </math>
! colspan="7" | <math display="inline">N</math> (Number of mesh points)
|-

! <math display="inline">\downarrow </math>
! 16
! 32
! 64
! 128
! 256
! 512
! 1024
|-

! 2<sup>−6</sup>
| 6.9037e−2
| 4.4673e−2
| 3.9152e−2
| 2.4284e−2
| 1.6478e−2
| 1.0510e−2
| 6.3683e−3
|-

! 2<sup>−7</sup>
| 7.1024e−2
| 4.5980e−2
| 3.3034e−2
| 2.7566e−2
| 1.6621e−2
| 1.0508e−2
| 6.3483e−3
|-

! 2<sup>−8</sup>
| 7.2001e−2
| 4.6628e−2
| 3.3467e−2
| 2.3775e−2
| 1.8534e−2
| 1.0523e−2
| 6.3774e−3
|-

! 2<sup>−9</sup>
| 7.2486e−2
| 4.6951e−2
| 3.3683e−2
| 2.3943e−2
| 1.6407e−2
| 1.1663e−2
| 6.3716e−3
|-

! 2<sup>−10</sup>
| 7.2727e−2
| 4.7112e−2
| 3.3790e−2
| 2.4026e−2
| 1.6488e−2
| 1.0447e−2
| 6.9885e−3
|-

! 2<sup>−11</sup>
| 7.2847e−2
| 4.7192e−2
| 3.3844e−2
| 2.4068e−2
| 1.6528e−2
| 1.0484e−2
| 6.3334e−3
|-

! 2<sup>−12</sup>
| 7.2907e−2
| 4.7232e−2
| 3.3871e−2
| 2.4089e−2
| 1.6548e−2
| 1.0502e−2
| 6.3508e−3
|-

! 2<sup>−13</sup>
| 7.2937e−2
| 4.7253e−2
| 3.3884e−2
| 2.4099e−2
| 1.6558e−2
| 1.0511e−2
| 6.3595e−3
|-

! 2<sup>−14</sup>
| 7.2952e−2
| 4.7263e−2
| 3.3891e−2
| 2.4104e−2
| 1.6563e−2
| 1.0516e−2
| 6.3638e−3
|-

! 2<sup>−15</sup>
| 7.2960e−2
| 4.7268e−2
| 3.3894e−2
| 2.4107e−2
| 1.6566e−2
| 1.0518e−2
| 6.3660e−3
|-

! 2<sup>−16</sup>
| 7.2964e−2
| 4.7270e−2
| 3.3896e−2
| 2.4108e−2
| 1.6567e−2
| 1.0519e−2
| 6.3671e−3
|-

! 2<sup>−17</sup>
| 7.2966e−2
| 4.7271e−2
| 3.3897e−2
| 2.4109e−2
| 1.6567e−2
| 1.0520e−2
| 6.3676e−3
|-

! 2<sup>−18</sup>
| 7.2967e−2
| 4.7272e−2
| 3.3897e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3679e−3
|-

! 2<sup>−19</sup>
| 7.2967e−2
| 4.7272e−2
| 3.3897e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3680e−3
|-

! 2<sup>−20</sup>
| 7.2967e−2
| 4.7272e−2
| 3.3898e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3681e−3
|-

! 2<sup>−21</sup>
| 7.2967e−2
| 4.7273e−2
| 3.3898e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3681e−3
|-

! 2<sup>−22</sup>
| 7.2967e−2
| 4.7273e−2
| 3.3898e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3682e−3
|-

! 2<sup>−23</sup>
| 7.2967e−2
| 4.7273e−2
| 3.3898e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3682e−3
|-

! 2<sup>−24</sup>
| 7.2967e−2
| 4.7273e−2
| 3.3898e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3682e−3
|-

! 2<sup>−25</sup>
| 7.2967e−2
| 4.7273e−2
| 3.3898e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3682e−3
|-

! 2<sup>−26</sup>
| 7.2967e−2
| 4.7273e−2
| 3.3898e−2
| 2.4109e−2
| 1.6568e−2
| 1.0520e−2
| 6.3681e−3
|-

! 2<sup>−27</sup>
| 7.2967e−2
| 4.7273e−2
| 3.3898e−2
| 2.4110e−2
| 1.6568e−2
| 1.0521e−2
| 6.3682e−3
|-

! <math display="inline">D^N</math>
| 7.2967e−2
| 4.7273e−2
| 3.9152e−2
| 2.7566e−2
| 1.8534e−2
| 1.1663e−2
| 6.9885e−3
|-

! <math display="inline">p^N</math>
| 6.2625e−1
| 2.7192e−1
| 5.0620e−1
| 5.7270e−1
| 6.6825e−1
| 7.3887e−1
| –
|}

<span id='f000005'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Nogué_166708860-1-s2.0-S1319516615000298-gr1.jpg|center|350px|Numerical solution of the problem stated in Example 8.1.]]


|-
| <span style="text-align: center; font-size: 75%;">

Fig. 1.

Numerical solution of the problem stated in [[#e000115|Example 8.1]].

</span>
|}

<span id='f000010'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Nogué_166708860-1-s2.0-S1319516615000298-gr2.jpg|center|368px|Maximum point wise error for the problem stated in Example 8.1.]]


|-
| <span style="text-align: center; font-size: 75%;">

Fig. 2.

Maximum point wise error for the problem stated in [[#e000115|Example 8.1]].

</span>
|}

<span id='e000120'></span>

==Example 8.2.                     ==

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
-\epsilon u^{{''}}(x)+(3+x^2)u^{{'}}(x)-u(x-1)=1\mbox{,}\quad x\in \Omega ^{-}\\
-\epsilon u^{{''}}(x)-(4+x)u^{{'}}(x)-u(x-1)=-1\mbox{,}\quad x\in \Omega ^+\\
u(x)=1\mbox{,}\quad x\in [-1,0]\mbox{,}\quad u(2)=2\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | (8.2)
|}

[[#t000010|Table 2]] presents the values of <math display="inline">D^N</math> and <math display="inline">p^N</math> for this problem. [[#f000015|Fig. 3]] and [[#f000020|Fig. 4]] represent the numerical solution and the maximum point wise error for this problem, respectively.

<span id='t000010'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 2.

Numerical results for the problem stated in [[#e000120|Example 8.2]].

|-

! <math display="inline">\epsilon </math>
! colspan="7" | <math display="inline">N</math> (Number of mesh points)
|-

! <math display="inline">\downarrow </math>
! 16
! 32
! 64
! 128
! 256
! 512
! 1024
|-

! 2<sup>−6</sup>
| 4.4334e−2
| 2.6901e−2
| 2.4542e−2
| 1.4287e−2
| 9.6205e−3
| 6.1830e−3
| 3.7665e−3
|-

! 2<sup>−7</sup>
| 4.5728e−2
| 2.7859e−2
| 1.9444e−2
| 1.6827e−2
| 9.7456e−3
| 6.1893e−3
| 3.7555e−3
|-

! 2<sup>−8</sup>
| 4.6412e−2
| 2.8332e−2
| 1.9796e−2
| 1.3927e−2
| 1.1337e−2
| 6.2057e−3
| 3.7820e−3
|-

! 2<sup>−9</sup>
| 4.6751e−2
| 2.8566e−2
| 1.9970e−2
| 1.4069e−2
| 9.5858e−3
| 7.1177e−3
| 3.7784e−3
|-

! 2<sup>−10</sup>
| 4.6920e−2
| 2.8683e−2
| 2.0057e−2
| 1.4140e−2
| 9.6544e−3
| 6.1463e−3
| 4.2869e−3
|-

! 2<sup>−11</sup>
| 4.7004e−2
| 2.8742e−2
| 2.0101e−2
| 1.4176e−2
| 9.6887e−3
| 6.1776e−3
| 3.7477e−3
|-

! 2<sup>−12</sup>
| 4.7046e−2
| 2.8771e−2
| 2.0122e−2
| 1.4193e−2
| 9.7058e−3
| 6.1932e−3
| 3.7624e−3
|-

! 2<sup>−13</sup>
| 4.7067e−2
| 2.8785e−2
| 2.0133e−2
| 1.4202e−2
| 9.7144e−3
| 6.2010e−3
| 3.7697e−3
|-

! 2<sup>−14</sup>
| 4.7077e−2
| 2.8792e−2
| 2.0139e−2
| 1.4207e−2
| 9.7186e−3
| 6.2049e−3
| 3.7734e−3
|-

! 2<sup>−15</sup>
| 4.7083e−2
| 2.8796e−2
| 2.0141e−2
| 1.4209e−2
| 9.7208e−3
| 6.2068e−3
| 3.7752e−3
|-

! 2<sup>−16</sup>
| 4.7085e−2
| 2.8798e−2
| 2.0143e−2
| 1.4210e−2
| 9.7219e−3
| 6.2078e−3
| 3.7761e−3
|-

! 2<sup>−17</sup>
| 4.7087e−2
| 2.8799e−2
| 2.0143e−2
| 1.4211e−2
| 9.7224e−3
| 6.2083e−3
| 3.7766e−3
|-

! 2<sup>−18</sup>
| 4.7087e−2
| 2.8799e−2
| 2.0144e−2
| 1.4211e−2
| 9.7227e−3
| 6.2086e−3
| 3.7768e−3
|-

! 2<sup>−19</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7228e−3
| 6.2087e−3
| 3.7769e−3
|-

! 2<sup>−20</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7229e−3
| 6.2087e−3
| 3.7770e−3
|-

! 2<sup>−21</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7229e−3
| 6.2088e−3
| 3.7770e−3
|-

! 2<sup>−22</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7229e−3
| 6.2088e−3
| 3.7770e−3
|-

! 2<sup>−23</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7229e−3
| 6.2088e−3
| 3.7770e−3
|-

! 2<sup>−24</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7229e−3
| 6.2088e−3
| 3.7770e−3
|-

! 2<sup>−25</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7229e−3
| 6.2088e−3
| 3.7770e−3
|-

! 2<sup>−26</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7229e−3
| 6.2088e−3
| 3.7770e−3
|-

! 2<sup>−27</sup>
| 4.7088e−2
| 2.8800e−2
| 2.0144e−2
| 1.4211e−2
| 9.7227e−3
| 6.2090e−3
| 3.7772e−3
|-

! <math display="inline">D^N</math>
| 4.7088e−2
| 2.8800e−2
| 2.4542e−2
| 1.6827e−2
| 1.1337e−2
| 7.1177e−3
| 4.2869e−3
|-

! <math display="inline">p^N</math>
| 7.0929e−1
| 2.3083e−1
| 5.4448e−1
| 5.6974e−1
| 6.7152e−1
| 7.3149e−1
| –
|}

<span id='f000015'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Nogué_166708860-1-s2.0-S1319516615000298-gr3.jpg|center|350px|Numerical solution of the problem stated in Example 8.2.]]


|-
| <span style="text-align: center; font-size: 75%;">

Fig. 3.

Numerical solution of the problem stated in [[#e000120|Example 8.2]].

</span>
|}

<span id='f000020'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Nogué_166708860-1-s2.0-S1319516615000298-gr4.jpg|center|368px|Maximum point wise error for the problem stated in Example 8.2.]]


|-
| <span style="text-align: center; font-size: 75%;">

Fig. 4.

Maximum point wise error for the problem stated in [[#e000120|Example 8.2]].

</span>
|}

==Example 8.3.                     ==

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace \begin{array}{c}
-\epsilon u^{{''}}(x)+(exp(x)+x^2)u^{{'}}(x)-u(x-1)=exp(x^2)\mbox{,}\quad x\in \Omega ^{-}\\
-\epsilon u^{{''}}(x)-(4+exp(-x))u^{{'}}(x)-u(x-1)=0\mbox{,}\quad x\in \Omega ^+\\
u(x)=1\mbox{,}\quad x\in [-1,0]\mbox{,}\quad u(2)=2\mbox{.}
\end{array}</math>
|}
| style="text-align: right;" | (8.3)
|}

[[#t000015|Table 3]] presents the values of <math display="inline">D^N</math> and <math display="inline">p^N</math> for this problem.

<span id='t000015'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 3.

Numerical results for the problem stated in Example 8.3.

|-

! <math display="inline">\epsilon </math>
! colspan="7" | <math display="inline">N</math> (Number of mesh points)
|-

! <math display="inline">\downarrow </math>
! 16
! 32
! 64
! 128
! 256
! 512
! 1024
|-

! 2<sup>−6</sup>
| 6.8150e−2
| 3.7767e−2
| 2.8472e−2
| 1.5380e−2
| 9.2391e−3
| 5.2438e−3
| 2.9335e−3
|-

! 2<sup>−7</sup>
| 6.9961e−2
| 3.8976e−2
| 2.3702e−2
| 1.7956e−2
| 9.3354e−3
| 5.3363e−3
| 3.0515e−3
|-

! 2<sup>−8</sup>
| 7.0863e−2
| 3.9595e−2
| 2.4127e−2
| 1.4829e−2
| 1.0949e−2
| 5.3842e−3
| 3.0680e−3
|-

! 2<sup>−9</sup>
| 7.1312e−2
| 3.9913e−2
| 2.4339e−2
| 1.4992e−2
| 9.2106e−3
| 6.2932e−3
| 3.0871e−3
|-

! 2<sup>−10</sup>
| 7.1537e−2
| 4.0072e−2
| 2.4446e−2
| 1.5074e−2
| 9.2781e−3
| 5.3106e−3
| 3.5991e−3
|-

! 2<sup>−11</sup>
| 7.1649e−2
| 4.0151e−2
| 2.4499e−2
| 1.5115e−2
| 9.3119e−3
| 5.3405e−3
| 3.0583e−3
|-

! 2<sup>−12</sup>
| 7.1705e−2
| 4.0191e−2
| 2.4525e−2
| 1.5135e−2
| 9.3288e−3
| 5.3554e−3
| 3.0725e−3
|-

! 2<sup>−13</sup>
| 7.1733e−2
| 4.0211e−2
| 2.4539e−2
| 1.5145e−2
| 9.3372e−3
| 5.3629e−3
| 3.0795e−3
|-

! 2<sup>−14</sup>
| 7.1747e−2
| 4.0221e−2
| 2.4545e−2
| 1.5150e−2
| 9.3415e−3
| 5.3666e−3
| 3.0831e−3
|-

! 2<sup>−15</sup>
| 7.1754e−2
| 4.0226e−2
| 2.4549e−2
| 1.5153e−2
| 9.3436e−3
| 5.3685e−3
| 3.0848e−3
|-

! 2<sup>−16</sup>
| 7.1758e−2
| 4.0228e−2
| 2.4550e−2
| 1.5154e−2
| 9.3446e−3
| 5.3694e−3
| 3.0857e−3
|-

! 2<sup>−17</sup>
| 7.1760e−2
| 4.0229e−2
| 2.4551e−2
| 1.5155e−2
| 9.3452e−3
| 5.3699e−3
| 3.0862e−3
|-

! 2<sup>−18</sup>
| 7.1760e−2
| 4.0230e−2
| 2.4552e−2
| 1.5155e−2
| 9.3454e−3
| 5.3701e−3
| 3.0864e−3
|-

! 2<sup>−19</sup>
| 7.1761e−2
| 4.0230e−2
| 2.4552e−2
| 1.5155e−2
| 9.3456e−3
| 5.3702e−3
| 3.0865e−3
|-

! 2<sup>−20</sup>
| 7.1761e−2
| 4.0230e−2
| 2.4552e−2
| 1.5155e−2
| 9.3456e−3
| 5.3703e−3
| 3.0865e−3
|-

! 2<sup>−21</sup>
| 7.1761e−2
| 4.0231e−2
| 2.4552e−2
| 1.5155e−2
| 9.3457e−3
| 5.3703e−3
| 3.0866e−3
|-

! 2<sup>−22</sup>
| 7.1761e−2
| 4.0231e−2
| 2.4552e−2
| 1.5155e−2
| 9.3457e−3
| 5.3703e−3
| 3.0866e−3
|-

! 2<sup>−23</sup>
| 7.1761e−2
| 4.0231e−2
| 2.4552e−2
| 1.5155e−2
| 9.3457e−3
| 5.3703e−3
| 3.0866e−3
|-

! 2<sup>−24</sup>
| 7.1761e−2
| 4.0231e−2
| 2.4552e−2
| 1.5155e−2
| 9.3457e−3
| 5.3704e−3
| 3.0866e−3
|-

! 2<sup>−25</sup>
| 7.1761e−2
| 4.0231e−2
| 2.4552e−2
| 1.5155e−2
| 9.3457e−3
| 5.3704e−3
| 3.0866e−3
|-

! 2<sup>−26</sup>
| 7.1761e−2
| 4.0231e−2
| 2.4552e−2
| 1.5155e−2
| 9.3457e−3
| 5.3703e−3
| 3.0866e−3
|-

! 2<sup>−27</sup>
| 7.1761e−2
| 4.0231e−2
| 2.4552e−2
| 1.5155e−2
| 9.3455e−3
| 5.3703e−3
| 3.0868e−3
|-

! <math display="inline">D^N</math>
| 7.1761e−2
| 4.0231e−2
| 2.8472e−2
| 1.7956e−2
| 1.0949e−2
| 6.2932e−3
| 3.5991e−3
|-

! <math display="inline">p^N</math>
| 8.3491e−1
| 4.9876e−1
| 6.6508e−1
| 7.1360e−1
| 7.9898e−1
| 8.0616e−1
| –
|}

==9. Discussion==

A BVP for one type of SPDDEs is considered. To obtain an approximate solution to this type of problem, an upwind finite difference scheme with piecewise linear interpolation on Shishkin mesh is presented. The method is shown to be of almost first order convergence. This is very much reflected on the numerical results ([[#t000005|Table 1]], [[#t000010|Table 2]] and [[#t000015|Table 3]]). Also [[#f000005|Fig. 1]] and [[#f000015|Fig. 3]] represent that the model problems stated in [[#e000115|Example 8.1]] and [[#e000120|Example 8.2]] exhibit strong interior layers at <math display="inline">x=1</math> and a weak boundary layer at <math display="inline">x=2</math>. [[#f000010|Fig. 2]] and [[#f000020|Fig. 4]] represent the maximum point wise error for the numerical solutions. Further these [[#f000010|Fig. 2]] and [[#f000020|Fig. 4]] represent the uniform convergence of the numerical method presented in this paper. The authors of  [[#br000035|[7]]] have considered second order ordinary differential equations with discontinuous convection coefficient with different signs on different subdomains. The solution to the problem considered in  [[#br000035|[7]]] exhibits strong interior layers at an interior point. Whereas the problem considered in this paper exhibits strong interior layers at <math display="inline">x=1</math> and weak boundary layer at <math display="inline">x=2</math> (see [[#e000045|Theorem 5.2]]). This is due to the presence of the delay term with the differential equation. Therefore, to accommodate these interior layers and boundary layer in numerical solution, the Shishkin mesh <math display="inline">\bar{\Omega }^N</math> has been constructed in Section  [[#s000035|6.1]]. In  [[#br000035|[7]]], the authors have suggested a uniform numerical method without interpolation, whereas the finite difference method with interpolation is needed in this paper, since the point <math display="inline">x_i-1,i>N/2</math> need not be a mesh point.

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