A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines

Abstract

We present an overlapping domain decomposition technique for solving the hypersingular integral equation on the sphere with spherical splines. We prove that the condition number of the additive Schwarz operator is bounded by O(H/δ), where H is the size of the coarse mesh and δ 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, a better bound O(1 + log 2 (H/δ)) is proved. The method is illustrated by numerical experiments on different point sets including those taken from magsat satellite data. Mathematics Subject Classification (2010) 65N55 · 65N30

A Posteriori Error Estimates for Hypersingular Integral Equation on Spheres with Spherical Splines

“…The linear system arising when solving this equation by using spherical splines is also ill-conditioned. However, preconditioners can be used to tackle this problem, see [36]. When solving the hypersingular integral equation (1.1) by using the Galerkin method with spherical splines associated with a regular and quasi-uniform spherical triangulation ∆, an a priori error estimate is proved as follows u − u ∆ H 1/2 (S) ≤ Ch…”

“…where τ (1) and τ (2) are spherical triangles in ∆ and the functions f 1 and f 2 are analytic for all x ∈ τ (1) and y ∈ τ (2) . For more details about the above evaluation, please refer to [36,38]. The right hand side F of the linear system (6.7) has entries given by…”

Efficient solvers for the shallow water equations on a sphere

“…Let ∆ H be a triangulation built on the set Y with the property H > h and let S H be the space of spherical splines defined on ∆ H . To obtain the decomposition (14) we first define a coarse space V 0 :=Ĩ h S H , whereĨ h is a quasi-interpolant operator [11, p. 425]. For each τ H j ∈ ∆ H where j = 1, .…”

“…Hence preconditioners are needed to overcome this problem. Recently additive Schwarz preconditioners were used to solve pseudodifferential equations on the unit sphere with rbfs [12,18] and with spherical splines [14,16]. Another kind of preconditioner, the alternate triangular preconditioner, was proposed by Samarskii [17] to solve the Poisson equation with a finite difference method on the unit square.…”