Analysis of approaches to spline interpolation of functions with large gradients in the boundary layer

The problem of cubic spline interpolation of functions with large gradients in the boundary layer is considered. It is assumed that the decomposition in the form of the sum of the regular and singular components is valid for the interpolated function. This decomposition is valid for the solution of a boundary value problem for a second order ordinary differential equation with a small parameter e at the highest derivative. An overview of approaches to the construction of splines, the error of which is uniform with respect to the small parameter e, is given. The approaches are based on the use of Shishkin and Bakhvalov meshes, which are thickened in the boundary layer region. An alternative approach based on a modification of a cubic spline with fitting to the singular component of the interpolated function is also considered. The comparison of the accuracy of the applied approaches with the performance of computational experiments is carried out.


Introduction
On the basis of boundary value problems for equations with a small parameter ε at higher derivatives, convective-diffusion processes with predominant convection are simulated. Solutions of such problems have large gradients in the boundary layer region, which affects the accuracy of classical difference schemes. For the numerical solution of such problems, special difference schemes are constructed, the error of which is uniform in the parameter ε. Two approaches are widely known for constructing ε-uniformly converging difference schemes: fitting the scheme to the boundary layer component [1] and using the classical difference schemes on grids, condensed in the boundary layer [2], [3].
The development of difference schemes for singularly perturbed problems has been the subject of the work of a number of authors. At the same time, the problem of developing spline interpolation methods for functions corresponding to the solution of singularly perturbed problems is also relevant. It is of interest to develop interpolation methods on grids that provide uniform convergence of difference schemes.
The spline theory has been extensively developed in the works of a number of authors. Note, for example, the monographs [4] [5], [6]. However, the development of splines for functions with a feature corresponding to the presence of a boundary layer is relevant. In accordance with [7] and other works, the application of polynomial interpolation formulas to functions with large gradients in the boundary layer can lead to errors of the order of O (1). Cubic splines are widely used to construct spline-collocation difference schemes for the numerical solution of singularly perturbed problems [8]. In this case, the error of splines between mesh nodes is not estimated.
In this paper, we will carry out a comparative analysis of the developed approaches to the construction of splines, having an error uniform over large gradients of the function in the boundary layer. For this, we will require from the splines that the error estimate is uniform with respect to a small parameter ε.
Notation. Let's set the grid We denote by S(Ω, k, 1) the space of polynomial splines of degree k of defect 1 [4] on the grid Ω. Throughout this work, by C and C j we mean positive constants independent of the parameter ε and the number of grid intervals N. In this case, we will limit different values by one constant C j , if it is clear from the text.

Preliminary information
Let the function u(x) be representable in the form: where for some constant C 1 the functions q(x) and Φ(x) are not explicitly given, α > 0, ε ∈ (0, 1]. According to (2), the regular component q(x) has derivatives bounded to the fourth order, and the boundary layer component Φ(x) has derivatives not uniformly bounded in the parameter ε. In accordance with [3], [9], a decomposition (1) with constraints (2) can be constructed for the solution of a singularly perturbed boundary value problem: where a 1 (x) ≥ α > 0, a 2 (x) ≥ 0, ε > 0, the functions a 1 (x), a 2 (x), f (x) are sufficiently smooth.
For small values of the parameter ε the solution of the problem (3) has a boundary layer region of large gradients at the boundary x = 0, to which the representation (1) corresponds. Let us study the problem of interpolation by a cubic spline of a function for which the decomposition (1) is valid.
Consider the case of a uniform grid Ω. Let S 3 (x, u) ∈ S(Ω, 3, 1) -cubic spline on the grid Ω, determined from the interpolation conditions In accordance with [10], the following theorem is true.
Theorem 1 In the case of a function u(x) of the form (1) and a uniform grid, there exists a constant C such that the estimate is valid. If in (1) Φ(x) = e −αx/ε , then it has place lower estimate From Theorem 1 it follows that in the case of a uniform grid, the error of interpolation by a cubic spline grows unboundedly if for a given N ε → 0. So, the problem of constructing a spline with ε-uniform error for a function of the form (1) is relevant.

Shishkin mesh
The analysis of the cubic spline error on the Shishkin mesh [3] was carried out in [10]. Let's set the Shishkin mesh Ω : Theorem 2 There are constants C and β > 0 that do not depend on ε, N , such that the error estimates hold It follows from (8) that this estimate is nonuniform with respect to the parameter ε, if On the intervals [x n , x n+1 ] for other values of n or for N ε ≥ 1, the error estimate is of the order of O((ln(N )/N ) 4 ) uniformly in the parameter ε.
The following theorem shows that the estimates (8) are unimprovable.
Theorem 3 Let be Φ(x) = e −x/ε . Then there are C, β > 0 that do not depend on ε, N such that the lower bounds hold In accordance with (9), on the Shishkin mesh, the spline error on some grid intervals grows indefinitely for ε → 0.
In accordance with [10], we modify the cubic spline on the Shishkin mesh by shifting one interpolation node. We putx N/2 = (x N/2 + x N/2+1 )/2,x n = x n , n ∈ [0, N/2 − 1] ∪ [N/2 + 1, N ]. Let S M (x, u) be an interpolation cubic spline determined from the conditions The only difference between the spline S M (x, u) and S 3 (x, u) is that the interpolation node x N/2 is replaced by the nodex N/2 . In this case, the nodes of the spline itself do not change and coincide with the nodes of Ω. In accordance with [11], the following theorem is true.
Theorem 4 Let the function u(x) have the representation (1), (2), Ω is the Shishkin mesh (6). Then, for some constant C, the following estimate holds Calculation of derivatives. It is known [4] that in the regular case, when the function u(x) has uniformly bounded derivatives, it is possible to calculate the derivatives based on the differentiation of the cubic spline, and the error estimates 3 (x, u)| ≤ Ch 4−j , j = 0, 1, 2 are valid, where h is the step of the uniform grid. Let us proceed to the analysis of this approach in the case when the function u(x) has large gradients and the decomposition (1) is valid for it. In accordance with [11], the following theorem is true.  (1), Ω is the Shishkin mesh (6), (7). Then, for some constant C, the error estimates are valid: In accordance with this theorem, relative estimates of the errors in the region of large gradients of the function and absolute estimates of the error outside the region of the boundary layer are obtained. Close estimates for the error in calculating derivatives on the Shishkin mesh are also valid for the cubic spline S 3 (x, u).

Bakhvalov mesh
The Bakhvalov mesh was developed in [2] for constructing an ε -uniformly converging difference scheme for the numerical solution of a singularly perturbed problem. It is of interest whether this mesh is applicable to function interpolation in the presence of a boundary layer. Let's define the Bakhvalov mesh of the interval [0, 1], based on the [2] approach with the [12] specification. We take into account the decomposition (1) for the interpolated function u(x).
The following theorem is proved in [13].
Theorem 6 There are constants C, C 1 and β > 0 that do not depend on ε, N such that for ε ≤ C/N the following estimate holds The resulting error estimate (12) is nonuniform with respect to the small parameter ε on mesh intervals when leaving the boundary layer.
In the case of ε ≥ C/N , in accordance with [14], the following theorem is true.
Due to the nonuniform convergence in ε of the cubic spline S 3 (x, u) according to estimates (12), by analogy with [10], we define a modified interpolation spline built on the basis of the shift of two interpolation nodes. With such a modification, the matrix of the system of equations for finding the coefficients in the expansion of the spline through the basis splines becomes strictly diagonally dominant. This leads to a ε-uniform estimate of the spline error.
So, put LetS 3 (x, u) be an interpolation cubic spline determined from the conditions Theorem 8 There are constants C, C 1 , independent of ε, N such that the following estimate Remark 1. We can assume that in Theorems 7 and 8 the constant C is the same. Otherwise, it is enough to take the minimum of these constants as C. Calculation of derivatives. In [15] estimates of the error in calculating derivatives on the basis of constructing a cubic spline S 3 (x, u) on a Bakhvalov mesh are obtained.
Theorem 9 Suppose that the function u(x) satisfies the representation (1), (2), Ω is a given Bakhvalov mesh. Then, for some constant C, the error estimate for x ∈ [x n−1 , x n ] is valid depending on the value of n : where β does not depend on ε.

Fitting a spline to a singular component
Consider a modification of a cubic spline based on fitting to the singular component responsible for large gradients of the function in the boundary layer. We will assume that the following decomposition is valid for the function u(x) : We assume that in the representation (13) the regular the component q(x) and the constant γ are not specified. Derivatives of function Φ(x) grow unboundedly near the boundary x = 0 if ε → 0. According to [16], the decomposition (13) is valid for the solution of singularly perturbed problem (3), with α = a 1 (0). In the general case, in accordance with [16], the derivatives of the function q(x), starting from the second, can grow with decreasing value of the parameter ε. This case requires additional investigation.
Let Ω be a uniform grid of the interval [0, 1] with nodes x n , n = 0, 1, . . . , N and step h. Let us define the space of L -splines exact on the singular component Φ(x) from (13): We define the interpolation L -spline S L (x, u) ∈ SL(Ω, 3, 1) from the conditions According to [17], the following theorem is true.
Theorem 10 Let the function u(x) have the decomposition (13). Then for the spline S L (x, u) the error estimates are valid In accordance with the estimate (14), for ε = 1, the error of the spline is of the order of O(h 4 ). This is consistent with the fact that as ε/h increases, the spline becomes cubic [17]. For ε ≤ h, the spline error becomes of the order of O(h 3 ). Calculation of derivatives. In [18], estimates of the error in calculating derivatives based on the differentiation of the spline S L (x, u) are obtained.
Theorem 11 Suppose that the function u(x) satisfies the decomposition (13), the grid is uniform. Then, for some constant C, the following error estimates hold
When calculating the derivatives of the function specified at the grid nodes, based on the differentiation of the spline, the following results are obtained. When the spline has the error that is not uniform in the parameter ε, then its application for calculating derivatives also leads to significant errors. Applying a cubic spline on the Shishkin and Bakhvalov meshes and the use of an exponential spline on a uniform grid gives the error in the calculation of derivatives uniform in the parameter ε. The greatest accuracy is obtained by using the Bakhvalov mesh. The errors of calculating the derivatives are shown in tables 6-8.

Conclusion
A comparison of the developed approaches to the interpolation of functions with large gradients in the exponential boundary layer is carried out. The use of a cubic spline on a uniform grid is unacceptable in the presence of an exponential boundary layer. The use of a cubic spline on the Shishkin and Bakhvalov meshes does not provide a uniform spline error in a small parameter. The error grows indefinitely at ε → 0 when leaving the region of high gradients. A shift of one interpolation node for the Shishkin mesh and two interpolation nodes for the Bakhvalov mesh leads to the fact that the error of a cubic spline becomes uniform in the parameter ε. In the case of a uniform grid, the use of an analog of a cubic spline, constructed on the basis of fitting to the singular component of the interpolated function, gives an error of the order O(N −3 ), uniform in the parameter ε. When ε 1/N, the exponential spline becomes cubic and the error becomes of the order of O(N −4 ). In accordance with the error estimates and the results of numerical experiments the cubic spline on the Bakhvalov mesh is more accurate.
Estimates of the error in calculating the derivatives of a function given at the grid nodes, based on spline interpolation, are given. It was found that if the error of the spline is uniform in the small parameter, then the error in calculating the derivatives is also uniform in the small parameter. The use of a uniform grid for calculating derivatives in the case of a cubic spline is unacceptable if the function has large gradients.