Radial basis functions for the solution of hypersingular operators on open surfaces

TL;DR

This paper presents a method using scaled radial basis functions and Lagrangian multipliers to solve hypersingular integral equations on open surfaces, demonstrating quasi-optimal convergence and competitive rates with boundary element methods.

Contribution

The paper introduces a novel approach combining scaled radial basis functions and Lagrangian multipliers for hypersingular operators on open surfaces, with proven convergence.

Findings

Method converges quasi-optimally.

Achieved convergence rates are close to boundary element schemes.

Numerical experiments confirm theoretical results.

Abstract

We analyze the approximation by radial basis functions of a hypersingular integral equation on an open surface. In order to accommodate the homogeneous essential boundary condition along the surface boundary, scaled radial basis functions on an extended surface and Lagrangian multipliers on the extension are used. We prove that our method converges quasi-optimally. Approximation results for scaled radial basis functions indicate that, for highly regular radial basis functions, the achieved convergence rates are close to the one of low-order conforming boundary element schemes. Numerical experiments confirm our conclusions.