A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines
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.
hypersingular integral equation, spherical splines, Ritz-Galerkin method
scientific article; zbMATH DE number 6006400