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The a priori known field behavior in the vicinity of simple, not grid conformal geometrical shapes is used to derive new approaches for the material averaging in the two-dimensional Finite Integration Technique. The problem of wave scattering at dielectric cylinders is investigated and the resulting absolute errors are well below the common facet-weighted averaging procedures. A mixed approach using normal and tangential field components separately is investigated. The reduced errors are presented and discussed for different permittivity contrasts.