Computing with Functions on Domains with Arbitrary Shapes

Abstract

We describe an approximation scheme and an implementation technique that enables numerical computations with functions defined on domains with an arbitrary shape. The scheme is spectrally accurate for smooth functions. The main advantage of the technique is that, unlike most spectral approximation schemes in higher dimensions, it is not limited to domains with tensor-product structure. The scheme itself is a discrete least squares approximation in a redundant set (a frame) that originates from a basis on a bounding box. The implementation technique consists of representing a domain by its characteristic function, i.e., the function that indicates whether or not a point belongs to the set. We show in a separate paper that the least squares approximation with N degrees of freedom can be solved in ${\mathscr {O}}(N^2\log ^2 N)$ operations for any domain that has non-trivial volume. The computational cost improves to ${\mathscr {O}}(N \log ^2 N)$ operations for domains that do have tensor-product structure. The scheme applies to domains even with fractal shapes, such as the Mandelbrot set, since such domains are defined precisely by their characteristic function.