Pseudodifferential Equations on the Sphere With Spherical Splines

Abstract

Spherical splines are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by using Galerkin method. We prove optimal convergence (in Sobolev norms) of the approximate solution by spherical splines to the exact solution. Our numerical results underlie the theoretical result.

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

“…For more details about the above evaluation, please refer to [15]. The right hand side of the linear system (2.21) has entries given by…”

“…Spherical splines (which were developed in [2][3][4] and have many properties in common with classical polynomial splines over planar triangulations) are another tool well suited for scattered data interpolation and approximation problems [15]. However, as proved in Proposition 2.6 in the next section and evidenced by the numerical results in Table 1, the Galerkin method with spherical splines yields a linear system which may also be ill-conditioned, though the ill-conditionedness is not as bad as with radial basis functions.…”

Efficient solvers for the shallow water equations on a sphere

“…However, finding the eigenvalues δ 1 and δ 2 is a more complicated problem than solving the original problem (15). Konovalov [8] showed that if {c l } are the iterates obtained by solving (15) with the steepest descend method and if…”

“…Numerical experiments show that the optimal value for κ(C(ω l )A) is κ(C(ω opt )A) , where ω opt = lim l→∞ωl . The obtained value ω opt is then used to solve (15) with the preconditioned conjugate gradient method with the optimal preconditioner C(ω opt ) . Table 2 gives a pseudocode for the proposed method to find ω opt .…”

“…Another possible way to deal with scattered data is to use spherical splines [1,3] (in the sense of Schumaker). Pham et al [15] designed a Galerkin method using spherical splines to approximate the solutions of pseudodifferential equations on the unit sphere.…”