On sinc discretization and banded preconditioning for linear third-order ordinary differential equations. (Q2889397)

The paper considers solving a linear third-order ODE boundary value problem. First, the ODE is discretized using a combination of the sinc-collocation and sinc-Galerkin methods. It is shown, that the discrete solution converges to the exact solution of the problem. The resulting system of linear algebraic equations, \(Aw=p\), is generally nonsymmetric and ill-conditioned. The matrix \(A\) is a linear combination of diagonal and Toeplitz matrices. Thus the algebraic system is solved by Krylov subspace methods (GMRES and BiCGSTAB) taking advantage of the fast matrix-vector multiplication. To accelerate the convergence, a class of banded preconditioning matrices for the considered problem is derived. Under some special assumptions on the coefficients of the ODE, it is proved that the eigenvalues of the preconditioned matrix lie in a rectangle on the complex plane. This rectangle is independent of the size of the algebraic system. Convergence is illustrated on two numerical experiments.