A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines
type: Item
instance of: scholarly article
title: A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines
zbMATH Open document ID: 1238.65133
author: Juan-Miguel Gracia
published in: Numerische Mathematik
publication date: 2012-02-14T00:00:00Z
review text: The paper is devoted to the approximate solution of hypersingular integral equations like \[ -\frac{1}{4\pi} \frac{\partial}{\partial v_x} \int\limits_s u(y) \frac{\partial}{\partial v_y} \frac{1}{|x-y|} d\sigma_y + \omega^2 \int\limits_s u d\sigma =f \] \(x \in S,\) based on the Ritz-Galerkin method with spherical splines. Here \(u(y)\) is an unknown function, \(\partial/\partial v_x\) is the normal derivate with respect to \(x,\) \(|.|\) denotes the Euclidean norm. The authors present an overlapping domain decomposition technique for solving the hypersingular integral equation on the sphere with spherical splines. They prove that the condition number of the additive Schwarz operator is bounded by \(O(H/\delta),\) where \(H\) is the size of the coarse mesh and \(\delta\) is the overlap size, which is chosen to be proportional to the size of the fine mesh. In the case that the degree of the splines is even, the better bound \(O(1+\log^2(H/\delta))\) is proved. The method is illustrated by numerical experiments.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45A05
zbMATH DE Number: 6006400
zbMATH Keywords: hypersingular integral equation, spherical splines, Ritz-Galerkin method
zbMATH Keywords: domain decomposition
zbMATH Keywords: numerical experiments
reviewed by: Ilia V. Boikov
MaRDI profile type: Publication
full work available at URL: s00211-011-0404-1
OpenAlex ID: W2016914753
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DOI: 10.1007/S00211-011-0404-1
description: scientific article; zbMATH DE number 6006400