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This paper describes a mimetic spectral element method on curvilinear grids applied to the Poisson equation. The Poisson equation is formulated in terms of differential forms. The spectral basis functions in which the differential forms are expressed lead to a metric free discrete representation of the gradient and the divergence operator. Using the fact that the pullback operator commutes with the wedge product and the exterior derivative leads to a mimetic spectral element formulation on curvilinear grids which displays exponential convergence and satisfies the divergence exactly. The robustness of the proposed scheme will be demonstrated for a sample problem for which exponential convergence is obtained. [ABSTRACT FROM AUTHOR]