  Analogues of the conjugate gradient method, MINRES, and GMRES are derived for
solving boundary value problems (BVPs) involving second-order differential
operators. Two challenges arise: imposing the boundary conditions on the
solution while building up a Krylov subspace, and guaranteeing convergence of
the Krylov-based method on unbounded operators. Our approach employs projection
operators to guarantee that the boundary conditions are satisfied, and we
develop an operator preconditioner that ensures that an approximate solution is
computed after a finite number of iterations. The developed Krylov methods are
practical iterative BVP solvers that are particularly efficient when a fast
operator-function product is available.
