The optimal 3-node preconditioner of the d2dx2 Fourier and Chebyshev spectral operators

It is shown in [G. Labrosse, The piecewise-linear Finite Volume scheme: the best known lowest-order preconditioner for the d2dx2 Chebyshev spectral operator, J. Comput. Phys. 228 (2009) 4491–4509] that the piecewise-linear Finite Volume (FV) approximation is the best preconditioner of the L[u]≡d2udx2=f Fourier and Chebyshev solvers compared to the corresponding 3-node Finite Difference (FD) and Finite Element (FE) approximations. The present paper reports about a 3-node preconditioner which, constructed from an appropriate linear combination of these FD, FE and FV approximations, is significantly better than the FV approximation. It is shown to be optimal.