Error estimates for harmonic and biharmonic interpolation splines with annular geometry

TL;DR

This paper provides error estimates for biharmonic interpolation splines in annular regions, extending one-dimensional spline techniques to higher dimensions and establishing bounds for key constants involved in the interpolation error analysis.

Contribution

It introduces a method to estimate errors for biharmonic polysplines in annuli, generalizing spline error bounds to higher-dimensional geometries with explicit constant bounds.

Findings

Derived bounds for the constant c(Ω) in annular regions.

Extended spline error estimation techniques to higher dimensions.

Provided explicit estimates for interpolation errors in annular geometries.

Abstract

The main result in this paper is an error estimate for interpolation biharmonic polysplines in an annulus $A\left( r_{1},r_{N}\right) $, with respect to a partition by concentric annular domains $A\left( r_{1} ,r_{2}\right) ,$ ...., $A\left( r_{N-1},r_{N}\right) ,$ for radii $0<r_{1}<....<r_{N}.$ The biharmonic polysplines interpolate a smooth function on the spheres $\left\vert x\right\vert =r_{j}$ for $j=1,...,N$ and satisfy natural boundary conditions for $\left\vert x\right\vert =r_{1}$ and $\left\vert x\right\vert =r_{N}.$ By analogy with a technique in one-dimensional spline theory established by C. de Boor, we base our proof on error estimates for harmonic interpolation splines with respect to the partition by the annuli $A\left( r_{j-1},r_{j}\right) $. For these estimates it is important to determine the smallest constant $c\left( \Omega\right) ,$ where $\Omega=A\left( r_{j-1},r_{j}\right) ,$ among all constants $c$ satisfying \[ \sup_{x\in\Omega}\left\vert f\left( x\right) \right\vert \leq c\sup _{x\in\Omega}\left\vert \Delta f\left( x\right) \right\vert \] for all $f\in C^{2}\left( \Omega\right) \cap C\left( \overline{\Omega }\right) $ vanishing on the boundary of the bounded domain $\Omega$ . In this paper we describe $c\left( \Omega\right) $ for an annulus $\Omega=A\left( r,R\right) $ and we will give the estimate \[ \min\{\frac{1}{2d},\frac{1}{8}\}\left( R-r\right) ^{2}\leq c\left( A\left( r,R\right) \right) \leq\max\{\frac{1}{2d},\frac{1}{8}\}\left( R-r\right) ^{2}% \] where $d$ is the dimension of the underlying space.