Convergence proofs are given for the projection based boundary element methods for the numerical solution of various Fourier problems in regions with smooth compact boundaries. Volterra integral equations of the 2nd kind are formulated with associated integral operators mapping the space of continuous functions on a compactum into itself. The compactness of these operators ia shown, yielding the error estimates in supremum norme for a wide class of projection based BEMs. Extensions of the error analysis to the initial -boundary value problems of convective heat conduction are also discussed.