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We present a Calder\'on preconditioning scheme for the symmetric formulationof the forward electroencephalographic (EEG) problem that cures both the densediscretization and the high-contrast breakdown. Unlike existing Calder\'onschemes presented for the EEG problem, it is refinement-free, that is, theelectrostatic integral operators are not discretized with basis functionsdefined on the barycentrically-refined dual mesh. In fact, in thepreconditioner, we reuse the original system matrix thus reducing computationalburden. Moreover, the proposed formulation gives rise to a symmetric,positive-definite system of linear equations, which allows the application ofthe conjugate gradient method, an iterative method that exhibits a smallercomputational cost compared to other Krylov subspace methods applicable tonon-symmetric problems. Numerical results corroborate the theoretical analysisand attest of the efficacy of the proposed preconditioning technique on bothcanonical and realistic scenarios.

On a Calder\'on preconditioner for the symmetric formulation of the electroencephalography forward problem without barycentric refinements