Exponential convergence of the hp-version of the composite Gauss-Legendre quadrature for integrals with endpoint singularities

Abstract

In this paper we propose and analyze an hp-version of the composite Gauss-Legendre quadrature rule which includes local subdivision of the integration domain and local variation of the number of quadrature nodes employed on each subinterval. We derive an a priori error estimate that is fully explicit in the local length of each subinterval, in the local number of quadrature nodes employed on each subinterval, and in the local regularity of the integrand. In particular, for integrands with singularities at one or both endpoints, we prove that exponential convergence can be achieved by using geometric partition of the integration domain and linearly increasing of the number of quadrature nodes in successive subintervals. Numerical experiments are provided to illustrate the theoretical results.