A note on an adaptive algorithm based on Chebyshev coefficients for two-point boundary value problems

Abstract

An adaptive version of an algorithm, first described by Greengard and Rokhlin, for numerical solution of two-point boundary value problems is proposed. The algorithm transforms two-point BVPs into integral equations, which are then solved by the Nystrom method using Chebyshev quadratures. The dense system of algebraic equations is solved in recursively in O(N) operations. The a posteriori node addition algorithm based on the size of Chebyshev coefficients of the solution approximations yields a robust method. The proposed approach combines the advantages of integral formulation and fast solution of dense linear systems with an automatic resolution of boundary and internal layers.