Multiview clustering (MC) aims to group samples using consistent and complementary information across various views. The subspace clustering, as a fundamental technique of MC, has attracted significant attention. In this paper, we propose a novel joint sparse self-representation learning model for MC, where a featured difference is the extraction of view-specific local information by introducing cardinality (i.e., $\ell_0$-norm) constraints instead of Graph-Laplacian regularization. Specifically, under each view, cardinality constraints directly restrict the samples used in the self-representation stage to extract reliable local and global structure information, while the low-rank constraint aids in revealing a global coherent structure in the consensus affinity matrix during merging. The attendant challenge is that Augmented Lagrange Method (ALM)-based alternating minimization algorithms cannot guarantee convergence when applied directly to our nonconvex, nonsmooth model, thus resulting in poor generalization ability. To address it, we develop an alternating quadratic penalty (AQP) method with global convergence, where two subproblems are iteratively solved by closed-form solutions. Empirical results on six standard datasets demonstrate the superiority of our model and AQP method, compared to eight state-of-the-art algorithms.