Abstract
Pseudodifferential equations on the unit sphere in $\R^n$, $n\ge 3$, are considered. The class of pseudodiffrential operators have long been used as a modern and powerful tool to tackle linear boundary-value problems. These equations arise in geophysics, where the sphere of interest is the earth. Efficient
solutions to these equations on the sphere become more demanding when given data are collected by satellites.
In this dissertation, firstly we solve these equations by using spherical radial basis functions. The use of these functions results in meshless methods, which have recently become more and more popular. In this dissertation, the collocation and Galerkin methods are used to solve pseudodifferential equations. From the
point of view of application, the collocation method is easier to implement, in
particular when the given data are scattered. However, it is well-known that the
collocation methods in general elicit a complicated error analysis. A salient
feature of our work is that error estimates for collocation methods are
obtained as a by-product of the analysis for the Galerkin method. This unified
error analysis is thanks to an observation that the collocation equation can be
viewed as a Galerkin equation, due to the reproducing kernel property of the
space in use.
Secondly, we solve these equations by using spherical splines with Galerkin
methods. Our main result is an optimal convergence rate of the approximation.
The key of the analysis is the approximation property of spherical
splines as a subset of Sobolev spaces. Since the pseudodifferential operators
to be studied can be of any order, it is necessary to obtain an approximation
property in Sobolev norms of any real order, negative and positive.
Solving pseudodifferential equations by using Galerkin methods with spherical
splines results, in general, in ill-conditioned matrix equations. To tackle this ill-conditionedness arising when solving two special pseudodifferential equations, the Laplace--Beltrami and hypersingular integral equations, we solve them by using a preconditioner which is defined by using the additive Schwarz method. Bounds for condition numbers of the preconditioned systems are established.