ON INTERPOLATION of FUNCTIONS with a BOUNDARY LAYER BY CUBIC SPLINES

Abstract

The problem of article is cubic spline-interpolation of functions having high gradient regions. It is shown that uniform grids are inefficient to be used. In case of piecewise-uniform grids, concentrated in the boundary layer, for cubic spline interpolation are announced asymptotically exact estimates on a class of functions with an exponential boundary layer. There are obtained results showing divergent in small parameter estimates and divergence of interpolation processes. The modified cubic spline with uniform in small parameter interpolation error is offered. The results of numerical experiments confirming theoretical estimates are given.

Keywords

• boundary layer
• singular perturbation
• cubic spline-interpolation
• error estimates

Introduction

Cubic splines are widely applied to smooth interpolation of functions. Such splines are investigated in [ 1 ], [ 2 ] and in many other works. However, according to [ 3 ], [ 4 ], application of the polynomial spline-interpolation to functions with large gradients in a boundary layer leads to essential errors of O(1) type. In [ 4 ] there was constructed a non-polynomial analogue of cubic spline, which is exact for the boundary layer component. Numerical experiments have shown the advantage in the accuracy of the constructed spline. However, the boundary layer component isn't always known, thus, in such case, there is no reasonable alternative for condensing a grid in the boundary layer. In this work a traditional cubic spline interpolation [ 2 ] on the piecewise uniform grid condensing in the boundary layer is investigated. There were obtained error estimates of interpolation which, however, aren't uniform in small parameter  . It is shown that an interpolation error of a boundary layer component might increase without limits while   0 , thus, we need to develop special methods of interpolation for such functions. Such a method of interpolation is also offered and investigated in this work. We shall pass it noting that the divergence of interpolation processes by cubic and parabolic splines on nonuniform grids was regarded in works [ 2 ], [ 5 ], [ 6 ] and some other works. However, the examples of divergence provided there had an artificial character or were implied with the help of Banach-Steinhaus theorem. In the present article we have shown divergence for functions describing solutions of a variety of applied problems. These results testify the need of development of universal high-order methods of smooth spline interpolation of functions on nonuniform grids and development of projective-grid methods of a high order for singular perturbed boundary value problems, because there is no need for application of the grid solution interpolation while using projective-grid methods.

Statement of the problem

Let us introduce the following notations. Let  : 0  x0  x1    xN  1 − partition of a segment [ 0,1 ] . Let S (, k,1) be a space of polynomial splines [ 2 ] of degree k and defect 1 on the grid  . We means that C,C j are positive constants independent from 

and a number of grid nodes. We write f  O(g) if f  C g and f  O* ( g ) , if f  O(g) and g  O( f ) , C[a,b] is a space of continuous functions with norm  Let a function u(x) have a form u(x)  q(x)  (x), x [ 0,1 ] ,

C[a,b]

. q( j) (x)  C , ( j) (x)  C ex / / j , 0  j  4 .

1 1 Let us investigate a problem of cubic spline-interpolation of the function (1).

Main results

First, let us look into the case of a uniform grid. Let N be a natural number,  is uniform grid with a step of H  1/ N and nodes xn , n  0,1,, N , partitioning of the interval [ 0,1 ] . Let g3 ( x, u)  S (,3,1) be an interpolation cubic spline on the grid  , defined by conditions: g3 ( xn , u)  u( xn ), 0  n  N , g3 '(0, u)  u'(0), g3 '(1, u)  u'(1). Theorem 1. In case of a uniform grid there shall be such a constant C for which the next estimate is correct: u(x)  g3 (x,u) (1) (2) u(x)  g3 (x,u) C[ 0,1 ]  C1 min (N )1, (N )4.

Next, according to [ 7 ] let us define the grid  steps with nodes xn , n  0,1,, N , and hn  h 

 N / 2 , n  1,, N , hn  H  1  , n  N  1,, N.

2 N / 2 2 In accordance with [ 7 ] let us define   min 1 , 4 ln N .

 2   (3) (4) (6) Due to [ 2 ] for interpolation cubic spline g3 ( x, u)  S (,3,1) the following error estimate is valid: g3 (x, u)  u(x)  5 384 u(4)

4 max hn .

C[ 0,1 ] n g3 (x, )  (x) C[xn ,xn1]  C5  N N5 e4ln(n4NN/ 2,0), Nn/ 2 Nn/ 2 N1 1 .

  Next Theorem shows us that estimates in (6) can not be improved.

Theorem 4.

Let ( x)  e x / .Then there are such constants C4 , C6 ,  1  0 , g3 ( x, )  ( x) independent from  , N that if   C4 N 1 , then

N 5 gm3 ( xn , u)  u( xn ) , n [0, N ] , gm3 ' (0, u)  u' (0), gm3 ' (1, u)  u' (1) . The only difference between gm3 ( x, u) and g 3 ( x, u) is that the interpolation node x N / 2 is set as xN / 2 . The spline nodes whereas are not subject to any changes and coincide with the  nodes.

Theorem 5. There are such independent from , N constants, namely 0  0, C that if  ln N   0 , it shall satisfy the following inequality : u(x)  gm3 (x,u) Therefore, according to theorems 2,5 if   O( N 1 ) and application of the interpolation spline the interpolation spline g3 ( x, u) if N 1  O( ) let as to obtain estimates (4), (7) uniformly in , N .

Results of Numerical Experiments

x 2 Let us define the following function:

 x u( x)  cos  e  , x [ 0,1 ]

.

Results of calculations are provided in the three following tables. Given in the tables below are the maximum errors of spline interpolation, calculated at nodes of the condensed grid, which is obtained from the initial grid by splitting every single grid interval into 10 parts. Table 1 contains interpolation errors for the traditional cubic spline on the uniform grid. Results confirm estimates of the Theorem 1 and an inadequacy of application of the uniform grid for small values of . Table 2 provides errors of a traditional cubic spline in Shishkin meshes. It follows from the tables that errors increase while  decreases and N is fixed. Results of Table 3 describing the errors of modified cubic spline, on the contrary, show uniform convergence, thus, theoretical conclusions are confirmed.

Acknowledgement

The work is supported by Russian Foundation of Basic Researches under Grant 1501-06584.

References

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