1 INTRODUCTION

Singularly perturbed problems are used to simulate various convective-diffusion processes with prevailing convection. It is known, e.g., from the work of A.M Il’in [1] that the use of classical difference schemes on a uniform mesh for the numerical solution of singularly perturbed problems leads to significant errors if the small parameter is comparable with the mesh size. In [1], to achieve the convergence of a difference scheme uniform in a small parameter, it was proposed to adjust the scheme to the singular component, which determined the main growth of the solution in the boundary layer. Another approach, based on mesh condensing in the boundary layer, was proposed by N.S. Bakhvalov [2]. Later, in the works of a number of authors, based on various approaches, mesh condensing in boundary layers were constructed, the use of which made it possible to ensure uniform convergence of the difference scheme, e.g., in [3, 4]. Wide application has been gained by the G.I. Shishkin mesh [5, 6].

Of significant interest is the question of interpolation of functions in the presence of regions of large gradients. It is well known that functions can be interpolated using Lagrange polynomials [7]. However, if the function has large gradients in the region of a boundary layer, the use of a Lagrange polynomial in the case of a uniform mesh can lead to errors on the order of \(O(1)\), if the small parameter is comparable with the mesh size [8]. It is of interest to consider the applicability of the Lagrange polynomial on meshes condensing in boundary layers, which are used in the construction of difference schemes.

In [9], the error of interpolation by a Lagrange polynomial of an arbitrary degree on a Shishkin mesh was estimated. For the Lagrange interpolation polynomial applied on non-intersecting subintervals of a mesh with \(k\) nodes, an error estimate of the order of \(O\left( {{{{(\ln (N){\text{/}}N)}}^{k}}} \right)\), uniform in the small parameter \(\varepsilon \), where \(N\) is the number of mesh steps, was obtained.

In the case of a Bakhvalov mesh [2], such a study was carried out in [10], where an \(\varepsilon \)-uniform error estimate of the order of \(O\left( {1{\text{/}}{{N}^{2}}} \right)\) for the piecewise-linear interpolation formula was obtained.

The aim of this work is to estimate the error of interpolation by a Lagrange polynomial with an arbitrary number of interpolation nodes on a Bakhvalov mesh. This research is new. The interpolation error will be estimated on the distinguished class of functions corresponding to the solution of a singularly perturbed problem in the case of an exponential boundary layer.

Thus, we assume that a function \(u(x)\) can be decomposed as

$$u(x) = p(x) + \Phi (x),\quad x \in [0,\;1],$$
(1.1)

where, for some constant \({{C}_{1}}\),

$$\left| {{{p}^{{(j)}}}(x)} \right| \leqslant {{C}_{1}},\quad \left| {{{\Phi }^{{(j)}}}(x)} \right| \leqslant \frac{{{{C}_{1}}}}{{{{\varepsilon }^{j}}}}{{e}^{{ - \alpha x/\varepsilon }}},\quad 0 \leqslant j \leqslant k,$$
(1.2)

the functions \(p(x)\) and \(\Phi (x)\) are not specified explicitly, \(\alpha > 0\), and \(\varepsilon \in (0,\;1]\). The coefficient \(\alpha \) is separated from zero, and the parameter \(k\) corresponds to the number of interpolation nodes. According to (1.2), the regular component \(p(x)\) has derivatives limited to a certain order, while the derivatives of the singular component \(\Phi (x)\) can increase without limit with a decrease in \(\varepsilon \).

In accordance with [5, 11], for a given \(k\), one can perform decomposition (1.1) with constraints (1.2) of the solution of a singularly perturbed boundary value problem

$$\varepsilon u''(x) + {{a}_{1}}(x)u'(x) - {{a}_{2}}(x)u(x) = f(x),\quad u(0) = A,\quad u(1) = B,$$
(1.3)

where \({{a}_{1}}(x) \geqslant \alpha > 0\), \({{a}_{2}}(x) \geqslant 0\), and \(\varepsilon > 0\) and the functions \({{a}_{1}}(x)\), \({{a}_{2}}(x)\), and \(f(x)\) are sufficiently smooth. For small \(\varepsilon \), the solution of problem (1.3) has a region of large gradients near the boundary \(x = 0,\) which corresponds to representation (1.1).

We introduce the following notation. Hereinafter, by \(C\) and \({{C}_{j}}\) we mean positive constants independent of the parameter \(\varepsilon \) and the number of mesh steps \(N\). One constant \({{C}_{j}}\) will be used to limit different quantities, if it is clear from the text. Let \({{L}_{{m,k}}}(u,x)\) be a Lagrange polynomial for a function \(u(x)\) with \(k\) interpolation nodes \({{x}_{m}},\; \ldots ,\;{{x}_{{m + k - 1}}}\) on the interval \(\left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\). Let \(K = 2(1 - \varepsilon )\) and \(u_{n}^{{(j)}} = {{u}^{{(j)}}}({{x}_{n}})\), \(j \geqslant 0\). We will write \(f = O(g)\) if \(\left| f \right| \leqslant C\left| g \right|\) and \(f = O{\text{*}}(g)\) if \(f = O(g)\) and \(g = O(f)\).

2 SPECIFYING A NONUNIFORM MESH

We specify the following mesh on the interval \([0,\;1]\):

$${{\Omega }^{h}} = \{ {{x}_{n}},\;n = 0,\;1,\; \ldots ,\;N,\;{{x}_{0}} = 0,\;{{x}_{N}} = 1,\;{{h}_{n}} = {{x}_{n}} - {{x}_{{n - 1}}}\} .$$

Based on [2], with the modification given in [3, 12], we specify the mesh nodes \({{x}_{n}} = g(n{\text{/}}N)\), defining the function \(g(t)\) as follows:

$$g(t) = - \frac{{k\varepsilon }}{\alpha }\ln [1 - 2(1 - \varepsilon )t],\quad 0 \leqslant t \leqslant \frac{1}{2},\quad \varepsilon \leqslant {{e}^{{ - 1}}},$$
(2.1)
$$g(t) = \sigma + (2t - 1)(1 - \sigma ),\quad 1{\text{/}}2 \leqslant t \leqslant 1,$$
(2.2)

where the parameter \(k\) corresponds to the number of interpolation nodes of the polynomial \({{L}_{{m,k}}}(u,x)\) specified below,

$$\sigma = \min \left\{ {\frac{1}{2}, - \frac{{k\varepsilon }}{\alpha }\ln \varepsilon } \right\}.$$
(2.3)

If \(\varepsilon > {{e}^{{ - 1}}}\), we take \(\sigma = 1{\text{/}}2\). For \(\sigma = 1{\text{/}}2\), we specify a uniform mesh \({{\Omega }^{h}}\) with steps \({{h}_{n}} = 1{\text{/}}N\).

Now we specify the mesh nodes for \(\sigma < 1{\text{/}}2\). Taking into account (2.1), we obtain

$${{x}_{n}} = - \frac{{k\varepsilon }}{\alpha }\ln [1 - 2(1 - \varepsilon )n{\text{/}}N],\quad n = 0,\;1,\; \ldots ,\;\frac{N}{2}.$$
(2.4)

Such nodes were introduced in [12]. It follows from (2.4) that

$${{h}_{n}} = \frac{{k\varepsilon }}{\alpha }\ln \left[ {1 + \frac{K}{{N - Kn}}} \right],\quad n = 1,\;2,\; \ldots ,\;N{\text{/}}2,\quad K = 2(1 - \varepsilon ).$$
(2.5)

It follows from (2.5) that the sequence of steps \(\{ {{h}_{n}}\} \), \(n = 1,\;2,\; \ldots ,\;N{\text{/}}2\), is strictly increasing and, for some constant \({{C}_{0}}\) and all \(n\), we have \({{h}_{n}} \leqslant {{C}_{0}}{\text{/}}N\). According to (2.5)

$${{h}_{{N/2}}} = \frac{{k\varepsilon }}{\alpha }\ln \left[ {1 + \frac{K}{{N\varepsilon }}} \right].$$
(2.6)

For \(n \geqslant N{\text{/}}2\), in accordance with (2.2), the mesh is uniform with steps \({{h}_{n}} = 2(1 - \sigma ){\text{/}}N\),

$${{x}_{n}} = \sigma + (2n{\text{/}}N - 1)(1 - \sigma ),\quad N{\text{/}}2 \leqslant n \leqslant N.$$
(2.7)

On the specified mesh, we estimate the error of the Lagrange interpolation polynomial \({{L}_{{m,k}}}(u,x)\) applied on nonintersecting intervals \(\left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\), \(m = 0,\;k - 1,\; \ldots ,\;N - k + 1\).

In the substantiation, we assume that \(\sigma < 1{\text{/}}2\), since, for \(\sigma = 1{\text{/}}2\), for some constant \(C\), we have \(\varepsilon \geqslant C\) and the derivatives of the function \(u(x)\) are \(\varepsilon \)-uniformly bounded. Then, we can apply the known estimates of the interpolation error and these estimates are not lower in order of accuracy than the estimates obtained in the case of \(\sigma < 1{\text{/}}2\).

3 ESTIMATION OF THE ERROR OF INTERPOLATION BY A LAGRANGE POLYNOMIAL

If \(N\) is a multiple of \(2(k - 1)\), then each interval \([{{x}_{m}},{{x}_{{m + k - 1}}}]\) falls entirely into the boundary layer region \([0,\sigma ]\) or will be outside it. The last such interval in the region \([0,\sigma ]\) will be the interval \(\left[ {{{x}_{{N/2 - k + 1}}},{{x}_{{N/2}}}} \right]\).

Consider a Lagrange polynomial

$${{L}_{{m,k}}}(u,x) = \sum\limits_{n = m}^{m + k - 1} {{{u}_{n}}} {{G}_{n}}(x),\quad {{G}_{n}}(x) = \prod\limits_{j = m,j \ne n}^{m + k - 1} {\frac{{x - {{x}_{j}}}}{{{{x}_{n}} - {{x}_{j}}}}} .$$
(3.1)

Theorem 1. Suppose that a function \(u(x)\) has representation (1.1), \(N\) is a multiple of \(2(k - 1)\), and nodes of the mesh \({{\Omega }^{h}}\) correspond to (2.3), (2.4), and (2.7). Then, for some constant \(C\), on the interval \(\left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\), for all \(m = 0,\;k - 1,\; \ldots ,\;N - k + 1\), depending on \(m\), we have the following error estimates:

$$\left| {{{L}_{{m,k}}}(u,x) - u(x)} \right| \leqslant \frac{C}{{{{N}^{k}}}},\quad x \in [{{x}_{m}},{{x}_{{m + k - 1}}}],\quad m + k - 1 < \frac{N}{2},$$
(3.2)
$$\left| {{{L}_{{m,k}}}(u,x) - u(x)} \right| \leqslant \frac{C}{{{{N}^{k}}}},\quad x \in [{{x}_{m}},{{x}_{{m + k - 1}}}],\quad m + k - 1 = \frac{N}{2},\quad \varepsilon \geqslant 1{\text{/}}N,$$
(3.3)
$$\left| {{{L}_{{N/2 - k + 1,k}}}(u,x) - u(x)} \right| \leqslant \frac{C}{{{{N}^{k}}}},\quad x \in [{{x}_{{N/2 - k + 1}}},{{x}_{{N/2 - 1}}}],$$
(3.4)
$$\left| {{{L}_{{N/2 - k + 1,k}}}(u,x) - u(x)} \right| \leqslant \frac{C}{{{{N}^{k}}}}{{\left( {\ln \left( {1 + \frac{1}{{N\varepsilon }}} \right)} \right)}^{{k - 1}}} + \frac{C}{{{{N}^{k}}}},\quad x \in [{{x}_{{N/2 - 1}}},{{x}_{{N/2}}}],$$
(3.5)
$$\left| {{{L}_{{m,k}}}(u,x) - u(x)} \right| \leqslant \frac{C}{{{{N}^{k}}}},\quad x \in [{{x}_{m}},{{x}_{{m + k - 1}}}],\quad m \geqslant \frac{N}{2}.$$
(3.6)

Proof. The polynomial \({{L}_{{m,k}}}(u,x)\) at all \(x \in \left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\) satisfies the error estimate [7]

$$\left| {{{L}_{{m,k}}}(u,x) - u(x)} \right| \leqslant \mathop {\max }\limits_{s \in [{{x}_{m}},{{x}_{{m + k - 1}}}]} \left| {{{u}^{{(k)}}}(s)} \right|\frac{{\left| {{{w}_{k}}(x)} \right|}}{{k!}},\quad {{w}_{k}}(x) = \prod\limits_{j = m}^{m + k - 1} {(x - {{x}_{j}})} .$$
(3.7)

Taking into account that \({{h}_{n}} \leqslant {{C}_{0}}{\text{/}}N\) for all \(n\;\) and estimate (1.2) for \(p(x)\), from (3.7), for some constant \(C\), we obtain

$$\left| {{{L}_{{m,k}}}(p,x) - p(x)} \right| \leqslant \frac{C}{{{{N}^{k}}}}.$$
(3.8)

It remains to estimate the interpolation error on the component \(\Phi (x)\) from (1.1).

Expanding the function \(\Phi (x)\) in a Taylor series, we obtain

$$\Phi (x) = {{P}_{k}}(x) + {{R}_{k}}(x),$$
(3.9)

where

$$\begin{gathered} {{P}_{k}}(x) = {{\Phi }_{m}} + (x - {{x}_{m}})\Phi _{m}^{'} + \frac{{{{{(x - {{x}_{m}})}}^{2}}}}{2}\Phi _{m}^{{''}} + \; \cdots \; + \frac{{{{{(x - {{x}_{m}})}}^{{k - 1}}}}}{{(k - 1)!}}\Phi _{m}^{{(k - 1)}}, \\ {{R}_{k}}(x) = \frac{1}{{(k - 1)!}}\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds. \\ \end{gathered} $$
(3.10)

In accordance with estimate (3.7), we have \({{L}_{{m,k}}}({{P}_{k}},x) - {{P}_{k}}(x) = 0\); therefore,

$${{L}_{{m,k}}}(\Phi ,x) - \Phi (x) = {{L}_{{m,k}}}({{R}_{k}},x) - {{R}_{k}}(x).$$
(3.11)

Taking into account (3.1), (3.10), and (3.11), we obtain

$$\begin{gathered} {{L}_{{m,k}}}(\Phi ,x) - \Phi (x) = \frac{1}{{(k - 1)!}}\sum\limits_{n = m}^{m + k - 1} {} \int\limits_{{{x}_{m}}}^{{{x}_{n}}} {{{{({{x}_{n}} - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds{{G}_{n}}(x) \\ - \;\frac{1}{{(k - 1)!}}\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds. \\ \end{gathered} $$
(3.12)

Let us estimate

$$\left| {\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds} \right|.$$
(3.13)

To do this, we consider different cases for the value of \(x\).

Case 1: \(x \in [{{x}_{m}},{{x}_{{m + k - 1}}}]\), \(m + k - 1 \leqslant N{\text{/}}2 - k + 1\). In accordance with (2.5) for \(m + k - 1 < N{\text{/}}2\), we have \({{h}_{m}} = O{\text{*}}({{h}_{{m + k - 1}}})\). Applying (1.2) and (2.5) in (3.13), for some constant \(C\), we obtain

$$\left| {\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds} \right| \leqslant \frac{C}{{{{\varepsilon }^{{k - 1}}}}}h_{m}^{{k - 1}}\left( {{{e}^{{ - \alpha {{x}_{m}}/\varepsilon }}} - {{e}^{{ - \alpha {{x}_{{m + k - 1}}}/\varepsilon }}}} \right).$$
(3.14)

Taking into account (2.4), for some constant \({{C}_{1}}\), we have

$$\begin{gathered} {{e}^{{ - \alpha {{x}_{m}}/\varepsilon }}} - {{e}^{{ - \alpha {{x}_{{m + k - 1}}}/\varepsilon }}} = (1 - 2(1 - \varepsilon )m{\text{/}}N{{)}^{k}} - {{(1 - 2(1 - \varepsilon )(m + k - 1){\text{/}}N)}^{k}} \\ \leqslant \;\frac{{{{C}_{1}}}}{N}{{(1 - 2(1 - \varepsilon )m{\text{/}}N)}^{{k - 1}}}. \\ \end{gathered} $$
(3.15)

We took into account the relations \({{b}^{k}} - {{a}^{k}} \leqslant k{{b}^{{k - 1}}}(b - a)\), \(0 < a < b < 1\), \(k > 1\).

Taking into account (2.5) and (3.15), from (3.14), for some constant \({{C}_{2}}\), we obtain

$$\left| {\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds} \right| \leqslant \frac{{{{C}_{2}}}}{{{{N}^{k}}}}\mathop {\ln }\nolimits^{k - 1} \left[ {1 + \frac{K}{{N - Km}}} \right]{{(N - Km)}^{{k - 1}}}.$$
(3.16)

From (3.16), we have

$$\left| {\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds} \right| \leqslant \frac{C}{{{{N}^{k}}}},\quad x \in [{{x}_{m}},{{x}_{{m + k - 1}}}],\quad m + k - 1 \leqslant N{\text{/}}2 - k + 1.$$
(3.17)

Case 2: \(x \in \left[ {{{x}_{{N/2 - k + 1}}},{{x}_{{N/2 - 1}}}} \right]\). Then, in inequality (3.16), we have \(\left| {x - s} \right| \leqslant C{{h}_{m}}\) and \(N - Km > K(k - 1) > 0\); therefore, we have the estimate

$$\left| {\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds} \right| \leqslant \frac{{{{C}_{3}}}}{{{{N}^{k}}}},\quad x \in \left[ {{{x}_{{N/2 - k + 1}}},{{x}_{{N/2 - 1}}}} \right].$$
(3.18)

Case 3: \(m = N{\text{/}}2 - k + 1\) and \(x \in \left[ {{{x}_{{N/2 - 1}}},{{x}_{{N/2}}}} \right]\). Then, \(\left| {x - s} \right| \leqslant C{{h}_{{N/2}}}\). Taking into account (2.6), by analogy with the previous cases, we obtain

$$\left| {\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds} \right| \leqslant \frac{C}{{{{N}^{k}}}}\mathop {\ln }\nolimits^{k - 1} \left[ {1 + \frac{K}{{N\varepsilon }}} \right]{{(N\varepsilon + K(k - 1))}^{{k - 1}}}.$$
(3.19)

From (3.19), for some constant \(C\), we obtain

$$\left| {\int\limits_{{{x}_{m}}}^x {{{{(x - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds} \right| \leqslant \frac{C}{{{{N}^{k}}}}\mathop {\ln }\nolimits^{k - 1} \left[ {1 + \frac{K}{{N\varepsilon }}} \right],\quad x \in \left[ {{{x}_{{N/2 - 1}}},{{x}_{{N/2}}}} \right].$$
(3.20)

Thus, (3.13) has been estimated depending on \(m\) and \(x\).

Let us now dwell on estimating the remaining terms in (3.12). Let us estimate the term in the sum from (3.12) for arbitrary n, considering various cases.

Case 1: \(x \in \left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\), \(m + k - 1 \leqslant N{\text{/}}2 - k + 1\). It follows from (2.5) that, for \(m \leqslant i,j \leqslant m + k - 1\), we have \({{h}_{i}} = O{\text{*}}({{h}_{j}})\). Therefore, for some constant \({{C}_{2}}\), we obtain

$$\left| {{{G}_{n}}(x)} \right| = \prod\limits_{j = m,j \ne n}^{m + k - 1} {\left| {\frac{{x - {{x}_{j}}}}{{{{x}_{n}} - {{x}_{j}}}}} \right|} \leqslant {{C}_{2}},\quad x \in \left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right].$$
(3.21)

Taking into account (3.21), by analogy with (3.17), for some constant \({{C}_{1}}\), we obtain

$$\left| {\frac{1}{{(k - 1)!}}\int\limits_{{{x}_{m}}}^{{{x}_{n}}} {{{{({{x}_{n}} - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds\prod\limits_{j = m,j \ne n}^{m + k - 1} {\frac{{x - {{x}_{j}}}}{{{{x}_{n}} - {{x}_{j}}}}} } \right| \leqslant \frac{{{{C}_{1}}}}{{{{N}^{k}}}}.$$
(3.22)

Case 2: \(x \in \left[ {{{x}_{{N/2 - k + 1}}},{{x}_{{N/2}}}} \right]\). We take into account that \({{h}_{i}} = O{\text{*}}({{h}_{j}})\) for \(N{\text{/}}2 - k + 1 \leqslant i,j < N{\text{/}}2\) and \({{h}_{{N/2 - 1}}} \ll {{h}_{{N/2}}}\) for small \(\varepsilon \).

For \(x,{{x}_{n}} \leqslant {{x}_{{N/2 - 1}}}\), we have

$$\prod\limits_{j = m,j \ne n}^{m + k - 1} {\left| {x - {{x}_{j}}} \right|} = O{\text{*}}\left( {h_{{m + 1}}^{{k - 2}}{{h}_{{N/2}}}} \right),\quad \prod\limits_{j = m,j \ne n}^{m + k - 1} {\left| {{{x}_{n}} - {{x}_{j}}} \right|} = O{\text{*}}\left( {h_{{m + 1}}^{{k - 2}}{{h}_{{N/2}}}} \right).$$

Taking into account these relations, we obtain estimate (3.22).

For \(x \leqslant {{x}_{{N/2 - 1}}}\), \(n = N{\text{/}}2\), we have

$$\prod\limits_{j = m,j \ne n}^{m + k - 1} {\left| {x - {{x}_{j}}} \right|} = O{\text{*}}\left( {h_{{m + 1}}^{{k - 1}}} \right),\quad \prod\limits_{j = m,j \ne n}^{m + k - 1} {\left| {{{x}_{n}} - {{x}_{j}}} \right|} = O{\text{*}}\left( {h_{{N/2}}^{{k - 1}}} \right).$$

Taking into account the inequality \(\left| {{{x}_{n}} - s} \right| \leqslant C{{h}_{{N/2}}}\), we find that estimate (3.22) is true.

For \(x \in \left[ {{{x}_{{N/2 - 1}}},{{x}_{{N/2}}}} \right]\), \(n < N{\text{/}}2\), we have

$$\left| {{{G}_{n}}(x)} \right| = \prod\limits_{j = m,j \ne n}^{m + k - 1} {\left| {\frac{{x - {{x}_{j}}}}{{{{x}_{n}} - {{x}_{j}}}}} \right|} \leqslant C{{\left( {{{h}_{{N/2}}}{\text{/}}{{h}_{m}}} \right)}^{{k - 2}}}.$$
(3.23)

Using (3.23), we obtain

$$\left| {\frac{1}{{(k - 1)!}}\int\limits_{{{x}_{m}}}^{{{x}_{n}}} {{{{({{x}_{n}} - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds\prod\limits_{j = m,j \ne n}^{m + k - 1} {\frac{{x - {{x}_{j}}}}{{{{x}_{n}} - {{x}_{j}}}}} } \right| \leqslant \frac{{{{C}_{1}}}}{{{{N}^{k}}}}\mathop {\ln }\nolimits^{k - 2} \left( {1 + \frac{K}{{N\varepsilon }}} \right).$$
(3.24)

For \(x \in \left[ {{{x}_{{N/2 - 1}}},{{x}_{{N/2}}}} \right]\), \(n = N{\text{/}}2\), we obtain

$$\left| {\frac{1}{{(k - 1)!}}\int\limits_{{{x}_{m}}}^{{{x}_{n}}} {{{{({{x}_{n}} - s)}}^{{k - 1}}}} {{\Phi }^{{(k)}}}(s)ds\prod\limits_{j = m,j \ne n}^{m + k - 1} {\frac{{x - {{x}_{j}}}}{{{{x}_{n}} - {{x}_{j}}}}} } \right| \leqslant \frac{{{{C}_{1}}}}{{{{N}^{k}}}}\mathop {\ln }\nolimits^{k - 1} \left( {1 + \frac{K}{{N\varepsilon }}} \right).$$
(3.25)

Case 3: \(m \geqslant N{\text{/}}2\). In view of (1.2), the derivatives of the function \(u(x)\) on the interval \(\left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\) are \(\varepsilon \)-uniformly bounded; therefore, in accordance with (3.7) and the estimate \({{h}_{n}} \leqslant C{\text{/}}N\), estimate (3.6) holds.

From relation (3.12) and estimates (3.8), (3.17), (3.18), (3.20), (3.22), (3.24), and (3.25), we obtain estimates (3.2)–(3.5). The theorem is proved.

4 STABILITY ESTIMATE FOR THE LAGRANGE POLYNOMIAL ON A BAKHVALOV MESH

Let us consider the question of the stability of a Lagrange polynomial \({{L}_{{m,k}}}(u,x)\) to a perturbation of \(u(x)\) at the nodes of a Bakhvalov mesh. Let \({{\tilde {u}}_{n}}\) be the perturbed value for \({{u}_{n}} = u({{x}_{n}})\). Then, in accordance with (3.1), we have

$$\left| {{{L}_{{m,k}}}(u,x) - {{L}_{{m,k}}}(\tilde {u},x)} \right| \leqslant \mathop {\max }\limits_{m \leqslant n \leqslant m + k - 1} \left| {{{u}_{n}} - {{{\tilde {u}}}_{n}}} \right|{{\Lambda }_{m}},$$
(4.1)

where \({{\Lambda }_{m}}\) is the Lebesgue constant [7]:

$${{\Lambda }_{m}} = \mathop {\max }\limits_{x \in \left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]} \sum\limits_{n = m}^{m + k - 1} {\left| {{{G}_{n}}(x)} \right|} .$$

Let \(m + k - 1 < N{\text{/}}2\). As shown above, in this case, estimate (3.21) holds. Therefore, \({{\Lambda }_{m}} \leqslant C\), \(C = k{{C}_{2}}\), where \({{C}_{2}}\) corresponds to (3.21). From (4.1), for \(x \in \left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\), we obtain a stability estimate

$$\left| {{{L}_{{m,k}}}(u,x) - {{L}_{{m,k}}}(\tilde {u},x)} \right| \leqslant C\mathop {\max }\limits_{m \leqslant n \leqslant m + k - 1} \left| {{{u}_{n}} - {{{\tilde {u}}}_{n}}} \right|.$$
(4.2)

Let \(m + k - 1 = N{\text{/}}2\). As shown above, the estimate of \({{G}_{n}}(x)\) is nonuniform in the parameter \(\varepsilon \) only if \(x \in \left[ {{{x}_{{N/2 - 1}}},{{x}_{{N/2}}}} \right]\), \(n < N{\text{/}}2\). In this case, estimate (3.23) holds. From (3.23), it follows that

$$\left| {{{G}_{n}}(x)} \right| \leqslant C\mathop {\ln }\nolimits^{k - 2} \left( {1 + \frac{1}{{N\varepsilon }}} \right).$$
(4.3)

Taking into account (4.1) and (4.3), for \(x \in \left[ {{{x}_{{N/2 - k + 1}}},{{x}_{{N/2}}}} \right]\), for some costant \({{C}_{1}}\), we obtain

$$\left| {{{L}_{{m,k}}}(u,x) - {{L}_{{m,k}}}(\tilde {u},x)} \right| \leqslant {{C}_{1}}\mathop {\max }\limits_{m \leqslant n \leqslant m + k - 1} \left| {{{u}_{n}} - {{{\tilde {u}}}_{n}}} \right|\mathop {\ln }\nolimits^{k - 2} \left( {1 + \frac{1}{{N\varepsilon }}} \right).$$
(4.4)

For \(m \geqslant N{\text{/}}2\), on the interval \(\left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\), the mesh is uniform; therefore, the stability estimate (4.2) holds.

Thus, we have obtained stability estimates (4.2) and (4.4). Estimate (4.4) for the interpolation interval \(\left[ {{{x}_{{N/2 - k + 1}}},{{x}_{{N/2}}}} \right]\) has a weak logarithmic dependence on the parameter \(\varepsilon \) and corresponds to estimate (3.5) for the error of interpolation by a Lagrange polynomial on this interval.

5 NEWTON–COTES FORMULAS ON A BAKHVALOV MESH

Of significant interest is the question of numerical integration of functions with large gradients. It was shown in [13] that, in the case of a uniform mesh, the use of composite Newton–Cotes formulas with two or three nodes in the basic formula when integrating functions of the form (1.1) can lead to errors on the order of \(O(h)\), where \(h\) is the mesh size. For example, with a decrease in the parameter \(\varepsilon \), the error of composite Simpson’s formula increases from order \(O\left( {{{h}^{4}}} \right)\) to \(O(h)\). Thus, there is a topical problem of constructing quadrature formulas for functions with a singularity corresponding to the presence of large gradients in the boundary layer region.

In [13], quadrature formulas with two and three nodes, exact on the boundary layer component of the integrable function, were constructed. In [14], a similar formula with four nodes was substantiated. In [15], a quadrature formula exact on the boundary layer component was studied in the general case when the basic quadrature formula contains \(k\) nodes. An error estimate of the order of \(O\left( {{{h}^{{k - 1}}}} \right)\), uniform in the boundary-layer component, was obtained.

Let us dwell on the application of the Newton–Cotes formulas on meshes condensing in the boundary layer. Thus, suppose that

$$I(u) = \int\limits_0^1 u (x)dx,$$
(5.1)

and the function \(u(x)\) can be decomposed in (1.1). Assume that \(N\) is multiple of \(2(k - 1)\) and specify

$${{I}_{{m,k}}}(u) = \int\limits_{{{x}_{m}}}^{{{x}_{{m + k - 1}}}} u (x)dx$$
(5.2)

for \(m = 0,\;k - 1,\; \ldots ,\;N - k + 1\). To calculate integral (5.2), we construct a Newton–Cotes formula with \(k\) nodes, using Lagrange polynomial (3.1):

$${{S}_{{m,k}}}(u) = \int\limits_{{{x}_{m}}}^{{{x}_{{m + k - 1}}}} {{{L}_{{m,k}}}} (u,x)dx.$$
(5.3)

Based on (5.3), we define a composite quadrature formula

$${{S}_{k}}(u) = \sum\limits_{m = 0,k - 1}^{N - k + 1} {{{S}_{{m,k}}}} (u).$$
(5.4)

First, we dwell on a Shishkin mesh [5], specified by the relations

$$\sigma = \min \left\{ {\frac{1}{2},\frac{{k\varepsilon }}{\alpha }\ln N} \right\},\quad {{h}_{n}} = \frac{{2\sigma }}{N},\quad n \leqslant \frac{N}{2};\quad {{h}_{n}} = \frac{{2(1 - \sigma )}}{N},\quad n > \frac{N}{2},$$
(5.5)

where the parameter \(k\) corresponds to the number of interpolation nodes of the polynomial \({{L}_{{m,k}}}(u,x)\).

In [9], for functions decomposed in (1.1) with constraints (1.2), the error of formula (5.4) on mesh (5.5) was estimated. The following error estimates were obtained:

$$\begin{gathered} \left| {I(u) - {{S}_{k}}(u)} \right| \leqslant \frac{C}{{{{N}^{k}}}}\left[ {1 + \varepsilon \mathop {\ln }\nolimits^{k + 1} N} \right]\quad {\text{for}}\quad \varepsilon < \frac{\alpha }{{2k\ln N}}, \\ \left| {I(u) - {{S}_{k}}(u)} \right| \leqslant \frac{C}{{{{N}^{k}}}}\min \left\{ {\frac{1}{{{{\varepsilon }^{k}}}},\mathop {\ln }\nolimits^k N} \right\}\quad {\text{for}}\quad \varepsilon \geqslant \frac{\alpha }{{2k\ln N}}. \\ \end{gathered} $$
(5.6)

In accordance with (5.6), the error of the composite quadrature formula on the Shishkin mesh is of the order of \(O\left( {{{N}^{{ - k}}}} \right)\) for \(\varepsilon = O(1)\) and for sufficiently small \(\varepsilon \). If we do not impose restrictions on the parameter \(\varepsilon ,\) the error is estimated by \(O\left( {{{{(\ln (N){\text{/}}N)}}^{k}}} \right)\).

Let us apply the estimates obtained in Theorem 1 to estimate the error of the Newton–Cotes formulas on a Bakhvalov mesh.

Theorem 2. Suppose that a function \(u(x)\) has representation (1.1) and \(N\) is a multiple of \(2(k - 1)\), and let nodes of the mesh \({{\Omega }^{h}}\) correspond to (2.3), (2.4), and (2.7). Then, for some constant \(C\), we have an error estimate

$$\left| {\int\limits_0^1 {u(x)} dx - {{S}_{k}}(u)} \right| \leqslant \frac{C}{{{{N}^{k}}}}.$$
(5.7)

Proof. First, we estimate the error of the formula for \({{S}_{{m,k}}}(u)\). From (5.2) and (5.3), we have

$$\left| {{{I}_{{m,k}}}(u) - {{S}_{{m,k}}}(u)} \right| \leqslant ({{x}_{{m + k - 1}}} - {{x}_{m}})\mathop {\max }\limits_{x \in [{{x}_{m}},{{x}_{{m + k - 1}}}]} \left| {u(x) - {{L}_{{m,k}}}(u,x)} \right|.$$
(5.8)

By Theorem 1,

$$\left| {u(x) - {{L}_{{m,k}}}(u,x)} \right| \leqslant \frac{C}{{{{N}^{k}}}},\quad m + k - 1 < \frac{N}{2}.$$

Then, from (5.8), we obtain

$$\left| {{{I}_{{m,k}}}(u) - {{S}_{{m,k}}}(u)} \right| \leqslant \frac{C}{{{{N}^{{k + 1}}}}},\quad m + k - 1 < \frac{N}{2}.$$
(5.9)

Let us dwell on the case of an interval \(\left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\) when \(m + k - 1 = N{\text{/}}2\). Then, in accordance with estimate (3.5), we have

$$\left| {{{L}_{{m,k}}}(u,x) - u(x)} \right| \leqslant \frac{C}{{{{N}^{k}}}}\mathop {\ln }\nolimits^{k - 1} \left( {1 + \frac{1}{{N\varepsilon }}} \right) + \frac{C}{{{{N}^{k}}}}.$$

Taking into account estimates (5.8) and (2.6), for some constant \({{C}_{1}}\), we obtain

$$\left| {{{I}_{{m,k}}}(u) - {{S}_{{m,k}}}(u)} \right| \leqslant \frac{{{{C}_{1}}\varepsilon }}{{{{N}^{k}}}}\mathop {\ln }\nolimits^k \left( {1 + \frac{1}{{N\varepsilon }}} \right) + \frac{{{{C}_{1}}}}{{{{N}^{{k + 1}}}}}.$$
(5.10)

Taking into accout that

$$\frac{{{{C}_{1}}\varepsilon }}{{{{N}^{k}}}}\mathop {\ln }\nolimits^k \left( {1 + \frac{1}{{N\varepsilon }}} \right) = \frac{{{{C}_{1}}}}{{{{N}^{{k + 1}}}}}(N\varepsilon )\mathop {\ln }\nolimits^k \left( {1 + \frac{1}{{N\varepsilon }}} \right),$$

from (5.10), we obtain

$$\left| {{{I}_{{m,k}}}(u) - {{S}_{{m,k}}}(u)} \right| \leqslant \frac{{{{C}_{2}}}}{{{{N}^{{k + 1}}}}},\quad m + k - 1 = \frac{N}{2}.$$
(5.11)

On the intervals \(\left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right]\) with \(m \geqslant N{\text{/}}2\), the derivatives of the functions \(u(x)\) are bounded uniformly in \(\varepsilon \); therefore, estimate (5.11) holds. Taking into account (5.4) and estimates (5.9) and (5.11), we obtain estimate (5.7). The theorem is proved.

Note that, according to estimate (5.7), on a Bakhvalov mesh, the order of accuracy of the composite Newton–Cotes formulas is the same as in the regular case when the integrated function has uniformly bounded derivatives. In this sense, estimate (5.7) is optimal.

6 RESULTS OF NUMERICAL EXPERIMENTS

For numerical experiments, we define the function

$$u(x) = \cos \frac{{\pi x}}{2} + {{e}^{{ - x/\varepsilon }}},\quad x \in [0,\;1],\quad \varepsilon \in (0,\;1].$$
(6.1)

According to [9], in the case of a function of the form (1.1) and Shishkin mesh (5.5), for some constant \(C\), for all \(m\), we have

$$\left| {u(x) - {{L}_{{m,k}}}(u,x)} \right| \leqslant C{{\left( {\frac{{\ln N}}{N}} \right)}^{k}},\quad x \in \left[ {{{x}_{m}},{{x}_{{m + k - 1}}}} \right].$$
(6.2)

Tables 1–3 present the error of interpolation on the interval \([0,\;1]\) by a Lagrange polynomial \({{L}_{{m,3}}}(u,x)\):

$${{\Delta }_{{N,\varepsilon }}} = \mathop {\max }\limits_{m,j} \left| {{{L}_{{m,3}}}(u,{{{\tilde {x}}}_{{m,j}}}) - u({{{\tilde {x}}}_{{m,j}}})} \right|$$

in the cases of a uniform, Shishkin, and Bakhvalov meshes, where \({{\tilde {x}}_{{m,j}}}\) are the nodes of the condensing meshes, obtained from the original mesh \({{\Omega }^{h}}\) by dividing each mesh interval \(\left[ {{{x}_{{n - 1}}},{{x}_{n}}} \right]\) into \(10\) equal parts. Tables 2 and 3 additionally present the calculated order of accuracy

$${{M}_{{N,\varepsilon }}} = \mathop {\log }\nolimits_2 \frac{{{{\Delta }_{{N,\varepsilon }}}}}{{{{\Delta }_{{2N,\varepsilon }}}}}.$$

In the tables, \(e - m\) means \({{10}^{{ - m}}}\).

Table 1.   Error of interpolation with three nodes on a uniform mesh
Full size table
Table 2.   Error of interpolation with three nodes on a Shishkin mesh
Full size table
Table 3.   Error of interpolation with three nodes on a Bakhvalov mesh
Full size table

Table 1 immediately implies the unacceptability of using a uniform mesh for \(\varepsilon \leqslant 1{\text{/}}N\). For \(\varepsilon = h\), the maximum error does not change with decreasing \(h = 1{\text{/}}N\).

A comparison of the results of Tables 2 and 3 shows that the use of a Bakhvalov mesh leads to more accurate results. This corresponds to an error estimate of order \(O\left( {1{\text{/}}{{N}^{3}}} \right)\) for a Bakhvalov mesh and estimate (6.2) for \(k = 3\) for a Shishkin mesh.

Table 4 presents the error \({{\Delta }_{{N,\varepsilon }}}\) and the calculated order of accuracy \({{M}_{{N,\varepsilon }}}\) in the case of a Lagrange polynomial \({{L}_{{m,4}}}(u,x)\) and a Bakhvalov mesh. The calculation results agree with the estimates of Theorem 1.

Table 4.   Error of interpolation with four nodes on a Bakhvalov mesh
Full size table

Let us dwell on the numerical analysis of the quadrature formulas used to calculate integral (5.1) using function (6.1) as an example.

Let us dwell on the case of the trapezoid formula applied on each interval \(\left[ {{{x}_{m}},{{x}_{{m + 1}}}} \right]\). Table 5 presents the error and the calculated order of accuracy of the corresponding composite trapezoid formula in the case of a Bakhvalov mesh, depending on \(\varepsilon \) and \(N\). The data of Table 5 are consistent with the second order of accuracy, which corresponds to estimate (5.7).

Table 5.   Error of the Newton–Cotes formula with two nodes on a Bakhvalov mesh
Full size table

Now consider the Newton–Cotes formula with three nodes on the interval \(\left[ {{{x}_{m}},{{x}_{{m + 2}}}} \right]\) of a nonuniform mesh:

$$\begin{gathered} {{S}_{{m,3}}}(u) = \frac{1}{6}\left[ {{{u}_{m}}\left( { - h_{{m + 2}}^{2}{\text{/}}{{h}_{{m + 1}}} + 2{{h}_{{m + 1}}} + {{h}_{{m + 2}}}} \right) + {{u}_{{m + 1}}}\frac{{{{{({{h}_{{m + 1}}} + {{h}_{{m + 2}}})}}^{3}}}}{{{{h}_{{m + 1}}}{{h}_{{m + 2}}}}} } \right. \\ \left. {\mathop + \limits_{_{{_{{_{{}}}}}}} \;{{u}_{{m + 2}}}\left( {2{{h}_{{m + 2}}} - h_{{m + 1}}^{2}{\text{/}}{{h}_{{m + 2}}} + {{h}_{{m + 1}}}} \right)} \right]. \\ \end{gathered} $$
(6.3)

Table 6 presents the error and the calculated order of accuracy of the composite formula for \({{S}_{3}}(u)\) corresponding to basic formula (6.3) on a Bakhvalov mesh. For \(\varepsilon = 1\), the mesh is uniform and formula (6.3) passes to Simpson’s formula. According to [7], composite Simpson’s formula has an increased fourth order of accuracy, which is confirmed by the calculation results. Consistency with estimate (5.7) is preserved, since, according to [7], the central node for a symmetric formula with an odd number of nodes can be considered as a double one. It follows from Table 6 that, with a decrease in \(\varepsilon \), on a Bakhvalov mesh, the order of accuracy close to the fourth is preserved.

Table 6.   Error of the Newton–Cotes formula with three nodes on a Bakhvalov mesh
Full size table

7 CONCLUSIONS

For the class of functions with large gradients in the region of an exponential boundary layer, estimates have been obtained for the error of interpolation by a Lagrange polynomial of an arbitrary degree on a Bakhvalov mesh. The resulting error estimates are uniform in the small parameter everywhere except the last mesh interval in the boundary layer region, where the weak logarithmic dependence on the small parameter is preserved. Based on the error estimates for a Lagrange polynomial, error estimates for the Newton–Cotes formulas on a Bakhvalov mesh have been obtained. These estimates are uniform in the small parameter. Calculation results consistent with the estimates obtained have been presented.