The Galerkin and the finite element methods are used to study the onset of the double-diffusive convective regime in a rectangular porous cavity. The two vertical walls of the cavity are subject to constant fluxes of heat and solute while the two horizontal ones are impermeable and adiabatic. The analysis deals with the particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity. For this situation, a steady rest state solution corresponding to a purely diffusive regime is possible. To demonstrate whether the solution is stable or unstable, a linear stability analysis is carried out to describe the oscillatory and the stationary instability in terms of the Lewis number, Le, normalized porosity, ε, and the enclosure aspect ratio, A. Using the Galerkin finite element method, it is shown that there exists a supercritical Rayleigh number, RsupTC, for the onset of the supercritical convection and an overstable Rayleigh number, RoverTC, at which overstability may arise. Furthermore, the overstable regime is shown to exist up to a critical Rayleigh number, RoscTC, at which the transition from the oscillatory to direct mode convection occurs. By using an analytical method based on the parallel flow approximation, the convective heat and mass transfer is studied. It is found that, below the supercritical Rayleigh number, RsupTC, there exists a subcritical Rayleigh number, RsubTC, at which a stable convective solution bifurcates from the rest state through finite-amplitude convection. In the range of the governing parameters considered in this study, a good agreement is observed between the analytical predictions and the finite element solution of the full governing equations. In addition, it is found that, for a given value of the governing parameters, the converged solution can be permanent or oscillatory, depending on the porous-medium porosity value, ε.