  We present a Calder\'on preconditioning scheme for the symmetric formulation
of the forward electroencephalographic (EEG) problem that cures both the dense
discretization and the high-contrast breakdown. Unlike existing Calder\'on
schemes presented for the EEG problem, it is refinement-free, that is, the
electrostatic integral operators are not discretized with basis functions
defined on the barycentrically-refined dual mesh. In fact, in the
preconditioner, we reuse the original system matrix thus reducing computational
burden. Moreover, the proposed formulation gives rise to a symmetric,
positive-definite system of linear equations, which allows the application of
the conjugate gradient method, an iterative method that exhibits a smaller
computational cost compared to other Krylov subspace methods applicable to
non-symmetric problems. Numerical results corroborate the theoretical analysis
and attest of the efficacy of the proposed preconditioning technique on both
canonical and realistic scenarios.
