Prof Miguel Pinãr (University of Granada Spain)

Orthogonal Sobolev polynomials and spectral methods for boundary value problems on the unit ball

Our main objective in this talk is to demonstrate how orthogonal Sobolev polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schrödinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of polynomials satisfying the boundary conditions on the sphere, and a basis of
mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is provided in terms of spherical harmonics and univariate orthogonal Sobolev polynomials. The connection formula between these orthogonal Sobolev polynomials and classical orthogonal polynomials on the sphere is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, numerical experiments were presented.