# A posteriori error estimates for hypersingular integral equation on spheres with spherical splines

## TL;DR

This paper develops and validates a posteriori error estimates for solving hypersingular integral equations on spheres using spherical splines, enabling adaptive mesh refinement to improve computational efficiency.

## Contribution

It introduces new a posteriori residual and hierarchical error bounds for spherical splines on the sphere, facilitating adaptive algorithms for hypersingular integral equations.

## Key findings

- Error estimates are proven and validated numerically.
- Adaptive refinement reduces computational complexity.
- Results demonstrate improved accuracy and efficiency.

## Abstract

A posteriori residual and hierarchical upper bounds for the error estimates were proved when solving the hypersingular integral equation on the unit sphere by using the Galerkin method with spherical splines. Based on these a posteriori error estimates, adaptive mesh refining procedures are used to reduce complexity and computational cost of the discrete problems. Numerical experiments illustrate our theoretical results.