Quoc Thong (2003-08). Approximation of linear partial differential equations on spheres. Doctoral Dissertation. Le Gia - Texas A&M University (TAMU) Scholar

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