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We propose a scheme for tensor field interpolation which is inspired by subdivision surfaces in computer graphics. The method applies to Cartesian tensors of all ranks and imposes smoothness on the interpolated field by constraining the divergence and curl of the tensor field. Applying the method involves only a sparse matrix-vector multiplication at each iteration. Results are presented for rank 1, 2 and 4 tensors. These examples demonstrate that the subdivision scheme can better preserve FA and interpolate rotations than some other interpolation methods.