Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface
type: Item
instance of: scholarly article
title: Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface
zbMATH Open document ID: 1382.65472
DOI: 10.1134/S0012266117090129
author: A. V. Setukha
author: A. V. Semenova
published in: Differential Equations
publication date: 2017-12-18T00:00:00Z
review text: The paper is concerned with the numerical solution of a linear hypersingular integral equation of the form \[ \int_{S} g(y) \frac{\partial^2 F \left( x-y \right)}{ \partial n_x \partial n_y} d \sigma_y = f\left(x\right), \quad x \in S, \quad F \left( x - y\right) = \frac{1}{\left| x-y \right|}, \] where \(S\) is a smooth closed surface and \(\partial / \partial n_x\) denotes the derivative in the direction of the outward unit normal to \(S\) at the point \(x\). The hypersingular integrals are interpreted in the sense of Hadamard's finite parts. The authors study a collocation method with basis functions that are piecewise linear on the triangulation of the surface \(S\) and they propose quadrature formulas for computing integrals. The uniform convergence of the quadrature formulas at the interpolation nodes is proved and under additional assumptions on the surface \(S\) the uniform error estimates of the proposed method at the interpolation nodes are derived.
reviewed by: Dana Černá
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E05
zbMATH DE Number: 6819865
zbMATH Keywords: collocation method
zbMATH Keywords: piecewise linear approximation
zbMATH Keywords: linear hypersingular integral equation
zbMATH Keywords: quadrature formulas
zbMATH Keywords: convergence
zbMATH Keywords: error estimates
Wikidata QID: Q115250864
MaRDI profile type: Publication
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full work available at URL: s0012266117090129
OpenAlex ID: W2762142622
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description: scientific article; zbMATH DE number 6819865