On sinc discretization method and block-tridiagonal preconditioning for second-order differential-algebraic equations

In this paper, we obtain a sinc discretization method for solving the boundary value problems of differential-algebraic equations (DAEs) and prove that the discrete solutions converge to the true solutions of the DAEs exponentially. In the process of presenting the solution, we find that the discrete solution is determined by a large and ill-conditioned linear system. To efficiently solve the linear system, we construct block-tridiagonal preconditioners for the iterative methods. Moreover, we derive the bounds for the eigenvalues of the preconditioned matrices. Numerical experiments are given to illustrate the effective performance and applicability of our method.