A fast uniformly accurate global numerical approximation to solution and scaled derivative of system of singularly perturbed problems with multiple diffusion parameters on generalized adaptive mesh

Abstract

The aim of this article is to analyze a robust numerical approximation to the solution along with the scaled first derivative (physically known as diffusive flux) of its components of a weakly coupled system of singularly perturbed convection-diffusion boundary-value-problems having diffusion parameters of different orders of magnitude. To accomplish this, we construct a generalized S-mesh, a general form of the piecewise-uniform Shishkin mesh, which appropriately resolves the overlapping boundary layers caused by the multiple diffusion parameters. For discretizing the governing system of equations, we confine our study to the upwind finite difference scheme. At first, we prove that the numerical solution converges uniformly in $C^0$-norm with faster convergence rate on the generalized S-mesh than the standard Shishkin mesh. Afterwards, we prove uniform convergence of the global numerical approximation in an appropriate weighted $C^1$-norm with same order of accuracy, which essentially establishes global accuracy to the numerical solution as well as the scaled discrete derivative. We confirm the theoretical findings by conducting extensive numerical experiments on two test examples for both equal and unequal values of diffusion parameters.

Appendices

Appendix A Proof of Lemma 6

Proof

$(\mathbb {G}_{\varepsilon _2, i+1}-\mathbb {G}_{\varepsilon _2, i})=-\frac{\uplambda h_{i+1}}{\varepsilon _2}\mathbb {G}_{\varepsilon _2, i+1}$. So we have

$$\begin{aligned}&\texttt{L}_{x, \varepsilon _1}^{N}{{{\textbf {G}}}}_{\varepsilon _2, i} \!=\!\varepsilon _1 \frac{2\uplambda }{\varepsilon _2 (h_i+h_{i+1})}(\mathbb {G}_{\varepsilon _2, i+1} \!\!-\!\mathbb {G}_{\varepsilon _2, i})\!+\!\texttt{p}_{11}(x_i)\frac{\uplambda }{\varepsilon _2}\mathbb {G}_{\varepsilon _2, i+1}\!\!+\!(\texttt{q}_{11}(x_i)\!+\!\texttt{q}_{12}(x_i))\mathbb {G}_{\varepsilon _2, i}\\&\ge \text {C}\frac{\uplambda }{\varepsilon _2}\mathbb {G}_{\varepsilon _2, i+1}\Bigg [\texttt{p}_{11}(x_i)-\frac{2\varepsilon _1 \uplambda h_{i+1}}{\varepsilon _2 (h_i+h_{i+1})}\Bigg ] \ge \frac{\text {C} \uplambda \mathbb {G}_{\varepsilon _2, i}}{\varepsilon _2 + \uplambda h_{i+1}} \ge \frac{\text {C} \mathbb {G}_{\varepsilon _2, i}}{\max \{\varepsilon _2, h_{i+1}\}}. \end{aligned}$$

Similarly, one can establish the other inequality for $\texttt{L}_{x, \varepsilon _2}^{N}\varvec{\mathbb {G}}_{\varepsilon _2, i}$. Also, $(\mathbb {G}_{\varepsilon _1, i+1}-\mathbb {G}_{\varepsilon _1, i})=-\frac{\uplambda h_{i+1}}{\varepsilon _1}\mathbb {G}_{\varepsilon _1, i+1}$. It implies that

$$\begin{aligned}&\texttt{L}_{x, \varepsilon _2}^{N}{{\textbf {G}}}_{\varepsilon _1, i}\!=\!\!-\varepsilon _2 \frac{2\uplambda }{\varepsilon _1 (h_i\!+\!h_{i\!+\!1})}(\mathbb {G}_{\varepsilon _1, i\!+\!1}\!\!-\!\mathbb {G}_{\varepsilon _1, i})\!-\!\texttt{p}_{22}(x_i)\frac{\uplambda }{\varepsilon _1}\mathbb {G}_{\varepsilon _1, i+1}\!\!+\!(\texttt{q}_{21}(x_i)\!-\!\texttt{q}_{22}(x_i))\mathbb {G}_{\varepsilon _1, i}\\&\ge \frac{\uplambda }{\varepsilon _1}\mathbb {G}_{\varepsilon _1, i+1}\Bigg [\frac{2\varepsilon _2 \uplambda h_{i+1}}{\varepsilon _1 (h_i+h_{i+1})}-\texttt{p}_{22}(x_i)\Bigg ]-\text {C}_3\mathbb {G}_{\varepsilon _1, i} \\&\ge \text {C}_4 \frac{\uplambda }{\varepsilon _1}\mathbb {G}_{\varepsilon _1, i+1}-\text {C}_3\mathbb {G}_{\varepsilon _1, i}, \quad \text {since }\varepsilon _1=o(\varepsilon _2), \text {C}_4\text { can be sufficiently large}\\&\ge \frac{\text {C} \uplambda \mathbb {G}_{\varepsilon _1, i}}{\varepsilon _1 + \uplambda h_{i+1}} \ge \frac{\text {C} \mathbb {G}_{\varepsilon _1, i}}{\max \{\varepsilon _1, h_{i+1}\}}. \end{aligned}$$

$\square$

Appendix B Proof of Lemma 7

Proof

• (i) For $1 \le i < N_1$, we have

  $$\begin{aligned} \mathbb {G}_{\varepsilon _1, i} \le e^{\frac{-\uplambda x_i}{\varepsilon _1 \!+\! \uplambda h_i}}=e^{\frac{-\uplambda x_i}{\varepsilon _1 \!+\! \uplambda \texttt{h}^1}} \!\le \! e^{\frac{-2i{N_1}^{-1}\ln ^{\Bbbk } N}{1+2N_{1}^{-1}\ln ^{\Bbbk } N}} = (\ln ^{\Bbbk -1} N)^{\frac{-2i{N_1}^{-1}}{1+2N_{1}^{-1}\ln ^{\Bbbk } N}} \le \text {C} (\ln ^{\Bbbk -1} N)^{{-2iN_{1}^{-1}}}. \end{aligned}$$

• (ii) For $\displaystyle \sum \nolimits _{j=1}^{n-1}N_j \le i < \sum \nolimits _{j=1}^{n}N_j, ~\forall ~n \in \{ 2, 3, ..., \Bbbk \}$, and proceeding as above, we get

  $$\begin{aligned} \mathbb {G}_{\varepsilon _1, i}&\le e^{\frac{-\uplambda x_i}{\varepsilon _1 + \uplambda h_i}} =e^{\frac{-\uplambda x_i}{\varepsilon _1 + \uplambda \texttt{h}^{n}}}\nonumber \\&\le (\ln ^{\Bbbk -n+1}N)^{\frac{-2\big (1-(i-\sum _{j=1}^{n-1}N_j)N_n^{-1}\big )}{1+2N_{n}^{-1}(\ln ^{n-1}N -\ln ^{n} N)}}(\ln ^{\Bbbk -n}N)^{\frac{-2\big (i-\sum _{j=1}^{n-1}N_j\big )N_n^{-1}}{1+2N_{n}^{-1}(\ln ^{n-1}N -\ln ^{n} N)}}, \nonumber \\&\le \text {C} (\ln ^{\Bbbk -n+1}N)^{-2\big (1-(i-\sum _{j=1}^{n-1}N_j){N_{n}^{-1}}\big )}(\ln ^{\Bbbk -n}N)^{{-2\big (i-\sum _{j=1}^{n-1}N_j\big )N_{n}^{-1}}}. \end{aligned}$$

  (B.1)

  Since, $\ln ^{\Bbbk -n+1}N<\ln ^{\Bbbk -n}N$, the inequality in (B.1) implies that

  $$\begin{aligned} \mathbb {G}_{\varepsilon _1, i}&\le {\text {C}(\ln ^{\Bbbk -n+1}N)^{-2\big (1-(i-\sum _{j=1}^{n-1}N_j)N_n^{-1}\big )} (\ln ^{\Bbbk -n+1}N)^{-2\big (i-\sum _{j=1}^{n-1}N_j\big )N_n^{-1}}}\nonumber \\ {}&\le {\text {C}(\ln ^{\Bbbk -n+1} N)^{-2}}\le \text {C}(\ln ^{\Bbbk -n+1} N)^{-1}. \end{aligned}$$

  (B.2)

  In particular, let $\displaystyle i=\sum \nolimits _{j=1}^{\Bbbk }N_j-1$. Then, from the inequality in (B.1) for $n=\Bbbk$, we have

  $$\begin{aligned} \mathbb {G}_{\varepsilon _1, i}&\le {\text {C}(\ln N)^{-2(1-(N_{\Bbbk }-1)N_{\Bbbk }^{-1})} N^{-2(N_{\Bbbk }-1)N_{\Bbbk }^{-1}}}\nonumber \\&= {\text {C}(\ln N)^{-2N_{\Bbbk }^{-1}} N^{-2+2N_{\Bbbk }^{-1}}}\nonumber \\&\le \text {C} N^{-2+2N_{\Bbbk }^{-1}}\le \text {C} N^{-1}. \end{aligned}$$

  (B.3)

• (iii) For $\displaystyle \sum \nolimits _{j=1}^{\Bbbk }N_j \le i < \sum \nolimits _{j=1}^{\Bbbk +1}N_j$, we have

  $$\begin{aligned} \mathbb {G}_{\varepsilon _2, i}&\le e^{\frac{-\uplambda x_i}{\varepsilon _2 + \uplambda h_i}}= e^{\frac{-\uplambda x_i}{\varepsilon _2 + \uplambda \texttt{h}^{\Bbbk +1}}}\\&\le e^{\frac{-2(i-\sum _{j=1}^{\Bbbk }N_j)N_{\Bbbk +1}^{-1}\ln ^{\Bbbk } N}{1+2N_{\Bbbk +1}^{-1}\ln ^{\Bbbk } N}} \\&= (\ln ^{\Bbbk -1} N)^{\frac{-2(i-\sum _{j=1}^{\Bbbk }N_j)N_{\Bbbk +1}^{-1}}{1+2N_{\Bbbk +1}^{-1}\ln ^{\Bbbk } N}} \\&\le \text {C} (\ln ^{\Bbbk -1} N)^{{-2(i-\sum _{j=1}^{\Bbbk }N_j)N_{\Bbbk +1}^{-1}}}. \end{aligned}$$

• (iv) For $\displaystyle \sum \nolimits _{j=1}^{\Bbbk +n-1}N_j \le i < \sum \nolimits _{j=1}^{\Bbbk +n}N_j, ~\forall ~n \in \{ 2, 3, ..., \Bbbk \},$

  $$\begin{aligned} \mathbb {G}_{\varepsilon _2, i}&\le e^{\frac{-\uplambda x_i}{\varepsilon _2 + \uplambda h_i}} =e^{\frac{-\uplambda x_i}{\varepsilon _2 + \uplambda H^{\Bbbk +n}}}\nonumber \\&\le (\ln ^{\Bbbk -n+1}N)^{\frac{-2\big (1-(i-\sum _{j=1}^{\Bbbk +n-1}N_j)N_{\Bbbk +n}^{-1}\big )}{1+2N_{\Bbbk +n}^{-1}(\ln ^{n-1}N -\ln ^{n} N)}}(\ln ^{\Bbbk -n}N)^{\frac{-2\big (i-\sum _{j=1}^{\Bbbk +n-1}N_j\big )N_{\Bbbk +n}^{-1}}{1+2N_{\Bbbk +n}^{-1}(\ln ^{n-1}N -\ln ^{n} N)}}, \nonumber \\&\le \text {C} (\ln ^{\Bbbk -n+1}N)^{-2\Big (1-(i-\sum _{j=1}^{\Bbbk +n-1}N_j){N_{\Bbbk +n}^{-1}}\Big )}(\ln ^{\Bbbk -n}N)^{{-2\big (i-\sum _{j=1}^{\Bbbk +n-1}N_j\big )N_{\Bbbk +n}^{-1}}}. \end{aligned}$$

  (B.4)

  Since, $\ln ^{\Bbbk -n+1}N<\ln ^{\Bbbk -n}N$, the inequality in (B.4) implies that

  $$\begin{aligned} \mathbb {G}_{\varepsilon _2, i}&\le {\text {C}(\ln ^{\Bbbk -n+1}N)^{-2\big (1-(i-\sum _{j=1}^{\Bbbk +n-1}N_j)N_{\Bbbk +n}^{-1}\big )} (\ln ^{\Bbbk -n+1}N)^{-2\big (i-\sum _{j=1}^{\Bbbk +n-1}N_j\big )N_{\Bbbk +n}^{-1}}}\nonumber \\ {}&\le \text {C}(\ln ^{\Bbbk -n+1} N)^{-2} \le \text {C}(\ln ^{\Bbbk -n+1} N)^{-1}. \end{aligned}$$

  (B.5)

  Let $\displaystyle i=\sum \nolimits _{j=1}^{2\Bbbk }N_j -1=N/2-1$. Then, from the inequality in (B.4) for $n=\Bbbk$, we have

  $$\begin{aligned} \mathbb {G}_{\varepsilon _2, i}&\le {\text {C}(\ln N)^{-2(1-(N_{2\Bbbk }-1)N_{2\Bbbk }^{-1})} N^{-2(N_{2\Bbbk }-1)N_{2\Bbbk }^{-1}}}\nonumber \\&= {\text {C}(\ln N)^{-2N_{2\Bbbk }^{-1}} N^{-2+2N_{2\Bbbk }^{-1}}}\nonumber \\&\le \text {C} N^{-2+2N_{2\Bbbk }^{-1}}\le \text {C} N^{-1}. \end{aligned}$$

  (B.6)

This completes the proof. $\square$

Appendix C Proof of Lemma 8

Proof

We define the truncation error associated with the smooth component $\varvec{S}$ by

$$\begin{aligned} \texttt{L}_{x, {\varepsilon _1}}^{N}(\varvec{S}-\varvec{s})(x_i) =\texttt{L}_{x, \varepsilon _1}\varvec{s}-\texttt{L}_{x, \varepsilon _1}^{N}\varvec{s} =\Big (\texttt{L}_{x, \varepsilon _1}-\texttt{L}_{x, \varepsilon _1}^{N}\Big )\varvec{s}. \end{aligned}$$

Then, by using the bounds of derivatives of $\varvec{s}$ given in Lemma  3, we get

$$\begin{aligned} \Bigg |\texttt{L}_{x, \varepsilon _1}^{N}(\varvec{S}-\varvec{s})(x_i)\Bigg |\le \frac{\varepsilon _1}{3}(h_i+h_{i+1})\Bigg |\Bigg |\frac{d^3 s_1}{d x^3}\Bigg |\Bigg |+\frac{h_{i+1}}{2}\texttt{p}_{11}(x_i)\Bigg |\Bigg |\frac{d^2 s_1}{d x^2}\Bigg |\Bigg | \le \text {C}N^{-1}. \end{aligned}$$

Similarly, we can prove that $\Big |\texttt{L}_{x, \varepsilon _2}^{N}(\varvec{S}-\varvec{s})(x_i)\Big |\le \text {C}N^{-1}.$ Now, we derive the desired bounds for the truncation error associated with the layer component $\varvec{R}$.

• (i) Let $x_i \in (0,\eta _{\varepsilon _1}^1).$ For $\displaystyle 1 \le i < N_1$, by using Lemma 3 and the method of [ Kellogg and Tsan (1978), Lemma 3.3] and since $\frac{\varepsilon _1}{\varepsilon _2} <1, \text {sinh}x \le x,$ we get

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _1}^{N}(\varvec{R}-\varvec{r})(x_i)\Big |&\le \text {C} \Bigg [\int _{x_{i-1}}^{x_{i+1}}(\varepsilon _1^{-2}\mathfrak {B}_{\varepsilon _1}(x)+ \varepsilon _2^{-2}\mathfrak {B}_{\varepsilon _2}(x))dx\Bigg ]\nonumber \\&\le \text {C} \Bigg [\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})\text {sinh}(\uplambda \texttt{h}^{1}/\varepsilon _1)+ \varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})\text {sinh}(\uplambda \texttt{h}^{1}/\varepsilon _2)\Bigg ]\nonumber \\&\le \text {C} \Bigg [\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})(\uplambda \texttt{h}^{1}/\varepsilon _1)+ \varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})(\uplambda \texttt{h}^{1}/\varepsilon _2)\Bigg ]\nonumber \\&\le \text {C} \Bigg [(\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})+\varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i}))N^{-1}\ln ^{\Bbbk } N\Bigg ]. \end{aligned}$$

  (C.1)

  Now, for $\displaystyle \sum \nolimits _{j=1}^{n-1}N_{j} \le i < \sum \nolimits _{j=1}^{n}N_{j}, \forall ~n \in \{2, ..., \Bbbk \},$ since $\eta _{\varepsilon _1}^1< \frac{\eta _{\varepsilon _2}^\Bbbk }{2} \Rightarrow \frac{\varepsilon _1}{\varepsilon _2} \ln N < \ln ^{\Bbbk }N$, we get $\frac{\varepsilon _1}{\varepsilon _2} (\ln ^n N-\ln ^{(n+1)}N) < \ln ^{\Bbbk }N,$ and proceeding as above we get

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _1}^{N}(\varvec{R}-\varvec{r})(x_i)\Big |&\le \text {C} \Bigg [\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})(\uplambda \texttt{h}^{n}/\varepsilon _1)+ \varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})(\uplambda \texttt{h}^{n}/\varepsilon _2)\Bigg ]\nonumber \\ {}&\le \text {C} \Bigg [\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})N^{-1}(\ln ^{\Bbbk -n+1}N-\ln ^{\Bbbk -n+2}N)\nonumber \\ {}&+\varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})(\varepsilon _1/\varepsilon _2)N^{-1}(\ln ^{\Bbbk -n+1}N-\ln ^{\Bbbk -n+2}N) \Bigg ]\nonumber \\&\le \text {C} \Bigg [\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})N^{-1}(\ln ^{\Bbbk -n+1}N-\ln ^{\Bbbk -n+2}N)\nonumber \\ {}&+\varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})N^{-1}\ln ^{\Bbbk }N \Bigg ]. \end{aligned}$$

  (C.2)

  Similarly, for $\displaystyle 1 \le i < N_1$, we have

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _2}^{N}(\varvec{R}-\varvec{r})(x_i)\Big | \le \text {C} \Bigg [(\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})+\varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i}))N^{-1}\ln ^{\Bbbk } N\Bigg ], \end{aligned}$$

  (C.3)

  and, for $\displaystyle \sum \nolimits _{j=1}^{n-1}N_{j} \le i < \sum \nolimits _{j=1}^{n}N_{j}, \forall ~n \in \{ 2, ..., \Bbbk \}$,

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _2}^{N}(\varvec{R}-\varvec{r})(x_i)\Big |&\le \text {C} \Bigg [\varepsilon _1^{-1}\mathfrak {B}_{\varepsilon _1}(x_{i})N^{-1}(\ln ^{\Bbbk -n+1}N-\ln ^{\Bbbk -n+2}N)\nonumber \\ {}&\quad \, +\varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})N^{-1}\ln ^{\Bbbk }N \Bigg ]. \end{aligned}$$

  (C.4)

• (ii) Let $x_i \in [\eta _{\varepsilon _1}^1, \eta _{\varepsilon _2}^1).$ Since $\varepsilon _1^{-2}\exp (-\uplambda x/\varepsilon _1) \le \varepsilon _2^{-2}\exp (-\uplambda x/\varepsilon _2)$, for $x > 2\varepsilon _1/\uplambda$, then we obtain, for $\displaystyle \sum \nolimits _{j=1}^{\Bbbk }N_{j} \le i < \sum \nolimits _{j=1}^{\Bbbk +1}N_{j},$ that

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _1}^{N}(\varvec{R}-\varvec{r})(x_i)\Big |&\le \text {C} \Big [\int _{x_{i-1}}^{x_{i+1}} \varepsilon _2^{-2}\mathfrak {B}_{\varepsilon _2}(x)dx\Big ]\nonumber \\&\le \text {C} \Big [\varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})(\uplambda \texttt{h}^{\Bbbk +1}/\varepsilon _2)\Big ]\nonumber \\&\le \text {C}\Big [ \varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_i)N^{-1}\ln ^{\Bbbk }N\Big ], \quad \uplambda \texttt{h}^{\Bbbk +1} \le \text {C}\varepsilon _2N^{-1}\ln ^{\Bbbk } N. \end{aligned}$$

  (C.5)

  Similarly, for $\displaystyle \sum \nolimits _{j=1}^{\Bbbk +n-1}N_{j} \le i < \sum \nolimits _{j=1}^{\Bbbk +n}N_{j}, \forall ~n \in \{2, ..., \Bbbk \},$

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _1}^{N}(\varvec{R}-\varvec{r})(x_i)\Big |&\le \text {C} \Big [\varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_{i})(\uplambda \texttt{h}^{\Bbbk +n}/\varepsilon _2)\Big ]\nonumber \\&\le \text {C}\Big [ \varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_i)N^{-1}(\ln ^{\Bbbk -n+1}N-\ln ^{\Bbbk -n+2}N) \Big ]. \end{aligned}$$

  (C.6)

  Apply same technique for $\displaystyle \sum \nolimits _{j=1}^{\Bbbk }N_{j} \le i < \sum \nolimits _{j=1}^{\Bbbk +1}N_{j},$ we have

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _2}^{N}(\varvec{R}-\varvec{r})(x_i)\Big |&\le \text {C}\Big [ \varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_i)N^{-1}\ln ^{\Bbbk }N\Big ], \end{aligned}$$

  (C.7)

  and for $\displaystyle \sum _{j=1}^{\Bbbk +n-1}N_{j} \le i < \sum _{j=1}^{\Bbbk +n}N_{j}, \forall ~n \in \{2, ..., \Bbbk \},$

  $$\begin{aligned} \Big |\texttt{L}_{x,\varepsilon _2}^{N}(\varvec{R}-\varvec{r})(x_i)\Big |\le \text {C}\Big [ \varepsilon _2^{-1}\mathfrak {B}_{\varepsilon _2}(x_i)N^{-1}(\ln ^{\Bbbk -n+1}N-\ln ^{\Bbbk -n+2}N)\Big ]. \end{aligned}$$

  (C.8)

Thus, we obtain the required truncation error bounds for $\varvec{R}$ from the inequalities (C.1)–(C.8). $\square$