Collocation methods for solving linear differential-algebraic boundary value problems

We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a = n - d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min{k + 1, 2k - 1} and superconvergence at mesh points of order 2k - 1 is shown. Finally, some numerical experiments illustrating these results are presented.