The piecewise-linear Finite Volume scheme: The best known lowest-order preconditioner for the Chebyshev spectral operator

The low-order (3-node) Finite Volume (FV) preconditioning technique is considered for its efficiency to spectrally solve the L [ u ] ≡ d 2 u dx 2 = f problem. The Fourier spectrum of the associated preconditioning operator is analytically determined and compared with the known Fourier spectra of the corresponding Finite Difference (FD) and Finite Element (FE) preconditioning operators. Moreover, following what was done in [P. Haldenwang, G. Labrosse, S. Abboudi, M.O. Deville, Chebyshev 3-d spectral and 2-d pseudospectral solvers for the Helmholtz equation, J. Comput. Phys. 55 (1984) 115–128] for the FD case, the FE and FV preconditioning operator Chebyshev spectra are analytically determined, first without reference to boundary conditions, and then confirmed when Dirichlet boundary conditions are imposed. The convergence of the Chebyshev towards the Fourier spectra is established. All this analysis leads to conclude that the best of the known lowest-order preconditioners is provided by the piecewise-linear Finite Volume scheme.