An adaptive partition of unity method for Chebyshev polynomial interpolation

摘要

For a function that is analytic on and around an interval, Chebyshevpolynomial interpolation provides spectral convergence. However, if thefunction has a singularity close to the interval, the rate of convergence isnear one. In these cases splitting the interval and using piecewiseinterpolation can accelerate convergence. Chebfun includes a splitting modethat finds an optimal splitting through recursive bisection, but the result hasno global smoothness unless conditions are imposed explicitly at thebreakpoints. An alternative is to split the domain into overlapping intervalsand use an infinitely smooth partition of unity to blend the local Chebyshevinterpolants. A simple divide-and-conquer algorithm similar to Chebfun'ssplitting mode can be used to find an overlapping splitting adapted to featuresof the function. The algorithm implicitly constructs the partition of unityover the subdomains. This technique is applied to explicitly given functions aswell as to the solutions of singularly perturbed boundary value problems.