Scalar and Vector Valued Radial Basis Functions

Abstract

Approximations of the differential operators such as laplacian operator, gradient operator, divergence operator, curl operator are important in numerical analysis. For example we can use these approximations for solving partial differential equations. Many times, their approximations on the sphere are required. In this thesis Wendland's radial basis functions are used for the approximations of differential operators. Firstly we study about the scalar radial basis functions and its interpolation method. We explained differential operators in terms of cartesian and spherical polar coordinates and scalar valued Wendland's radial basis functions which are used for the approximations of these operators on the sphere. Based of these radial basis functions, the approximations are computed. Next we considered divergence-free matrix valued radial basis functions generated by compactly supported scalar Wendland's radial basis functions.