Numerical Methods for Partial Differential Equations Seminar
Wednesday, November 29, 2017 at 4:30pm to 5:30pm
SPEAKER: Geoffrey Vasil (University of Sydney)
TITLE: Flexible tensor calculus on domains with coordinate singularities
On a two-sphere (for example), not all smooth functions of latitude and longitude represent non-singular physical quantities; the concept of longitude breaks down at the north and south poles. Truly well-behaved functions must contain special relationships between otherwise independent variables near coordinate singularities. In applications, capturing these dependencies can become numerically very challenging. Fortunately, the famous Spherical Harmonic functions perfectly satisfy the special behaviour (for scalar functions) near the poles and can be used as a complete spectral basis. But what happens for vectors, and higher-order tensors? What happens on the unit disk? What about the coordinate singularity at the centre of the 3D ball? Or, what about the corners of triangles and tetrahedra?
This talk will discuss how to represents functions, vectors, and arbitrary tensors on domains with coordinate singularities in an accurate and numerically efficient way. The solution is based on the remarkable properties of Jacobi polynomials; which include Legendre and Chebyshev polynomials as special cases, and Hermite and Laguerre as limiting cases. Using Jacobi polynomials, it is possible to construct basis sets that account for a wide range of singular behaviour. These bases inherit simple and sparse differentiation and multiplication formula from the algebra of Jacobi polynomials, which allows for the efficient solution of PDEs.