We present a new geometry-based numerical method that converges on grids with curved (i.e., non-planar) faces. Notably, it employs only one degree of freedom (DoF) per curved face. This is achieved by introducing a novel generalized dual grid, which is not constructed via barycentric subdivision, as in standard discrete geometric approaches, but is instead derived directly from the geometry of the curved faces. Crucially, the generalized dual grid is used in the numerical scheme in the same way as the standard barycentric one. As a result, all the foundational principles of geometric methods remain applicable to our new numerical approach.
