The theory of interpolation and approximation of solutions to differential and integral equations on spheres has attracted considerable interest in recent years; it has also been applied fruitfully in fields such as physical geodesy, potential theory, oceanography, and meteorology. In this dissertation we study the approximation of linear partial differential equations on spheres, namely a class of elliptic partial differential equations and the heat equation on the unit sphere. The shifts of a spherical basis function are used to construct the approximate solution. In the elliptic case, both the finite element method and the collocation method are discussed. In the heat equation, only the collocation method is considered. Error estimates in the supremum norms and the Sobolev norms are obtained when certain regularity conditions are imposed on the spherical basis functions.
The theory of interpolation and approximation of solutions to
differential and integral equations on spheres has attracted
considerable interest in recent years; it has also been applied
fruitfully in fields such as physical geodesy, potential theory,
oceanography, and meteorology.
In this dissertation we study the approximation of linear
partial differential equations on spheres, namely a class of
elliptic partial differential equations
and the heat equation on the unit sphere.
The shifts of a spherical basis
function are used to construct the approximate solution. In the
elliptic case, both the finite element method and the collocation method
are discussed. In the heat equation, only the collocation method is
considered. Error estimates in the supremum norms and the Sobolev norms
are obtained when certain regularity conditions are imposed on
