The Nonconvergence of $$h$$h-Refinement in Prolate Elements

• Authors: John P. Boyd;Gregor Gassner;Burhan A. Sadiq
• Venue: Journal of Scientific Computing
• Year: 2013

Prolate elements are a "plug-compatible" modification of spectral elements in which Legendre polynomials are replaced by prolate spheroidal wave functions of order zero. Prolate functions contain a"bandwidth parameter" $$c \ge 0 $$ c 驴 0 whose value is crucial to numerical performance; the prolate functions reduce to Legendre polynomials for $$c\,=\,0$$ c = 0 . We show that the optimal bandwidth parameter $$c$$ c not only depends on the number of prolate modes per element $$N$$ N , but also on the element widths $$h$$ h . We prove that prolate elements lack $$h$$ h -convergence for fixed $$c$$ c in the sense that the error does not go to zero as the element size $$h$$ h is made smaller and smaller. Furthermore, the theoretical predictions that Chebyshev and Legendre polynomials require $$\pi $$ 驴 degrees of freedom per wavelength to resolve sinusoidal functions while prolate series need only 2 degrees of freedom per wavelength are asymptotic limits as $$N \rightarrow \infty $$ N 驴 驴 ; we investigate the rather different behavior when $$N \sim O(4-10)$$ N ~ O ( 4 驴 10 ) as appropriate for spectral elements and prolate elements. On the other hand, our investigations show that there are certain combinations of $$N,\,h$$ N , h and $$c0$$ c 0 where a prolate basis clearly outperforms the Legendre polynomial approximation.