A Collocation–Galerkin Method for the Two Point Boundary Value Problem Using Continuous Piecewise Polynomial Spaces

A collocation–Galerkin method for the two point boundary value problem based on continuous piecewise polynomial spaces is defined where the collocation points are the roots of a Jacobi polynomial. An existence and uniqueness theorem is derived by means of a related variational procedure that is defined using a semi-discrete innerproduct. Optimal rates of convergence are established and O(h²ʳ ) order of superconvergence at the nodes is obtained. Additional approximations to the solution and its derivatives that obtain O(h²ʳ ) order of superconvergence at any point are described.