Igor Tominec (Stockholm University): Basis functions in finite difference methods

Basis functions are central to many numerical methods for solving PDEs. For example, the standard finite element methods use globally defined Lagrange basis functions with compact support. This enables weak formulations and a functional analysis framework for theoretical understanding of the method. In contrast, finite difference (FD) methods do not use globally defined basis functions. Instead, they approximate derivatives using differentiation weights derived from truncated Taylor expansions at each grid point.

In this talk, I will present a different viewpoint on FD methods, covering both the standard grid-based formulation and meshfree approaches based on radial basis function methods. I will discuss: (i) how to define global Lagrange basis functions with compact support, of arbitrary approximation order, (ii) the regularity of the resulting FD function spaces, and (iii) how this perspective aids in understanding the stability of FD methods for hyperbolic PDEs in the context of energy estimates.