Linear mixed model (LMM) has been extensively applied for hyperspectral compressive sensing (CS) in recent years. However, the error introduced by LMM that limits the reconstruction performance has not been given full consideration. We propose an algorithm for hyperspectral CS based on LMM under the assumption of known endmembers. At the sampling stage, only spectral compressive sampling is carried out to keep the abundance information as much as possible. At the reconstruction stage, the proposed algorithm estimates abundance by using linear unmixing from the spectral observed data. Moreover, the model error introduced by LMM is explored; a joint convex optimization scheme for estimation of both abundance and model error is established and solved by the alternating iteration approach to achieve the optimal reconstruction. Experimental results on a real hyperspectral dataset demonstrate that the proposed algorithm significantly outperforms the other state-of-the-art hyperspectral CS algorithms.