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This paper presents a novel jointly sparse signal reconstruction algorithm for the DOA estimation problem, aiming to achieve faster convergence rate and better estimation accuracy compared to existing l2,1-norm minimization approaches. The proposed greedy block coordinate descent (GBCD) algorithm shares similarity with the standard block coordinate descent method for l2,1-norm minimization, but adopts a greedy block selection rule which gives preference to sparsity. Although greedy, the proposed algorithm is proved to also have global convergence in this paper. Through theoretical analysis we demonstrate its stability in the sense that all nonzero supports found by the proposed algorithm are the actual ones under certain conditions. Last, we move forward to propose a weighted form of the block selection rule based on the MUSIC prior. The refinement greatly improves the estimation accuracy especially when two point sources are closely spaced. Numerical experiments show that the proposed GBCD algorithm has several notable advantages over the existing DOA estimation methods, such as fast convergence rate, accurate reconstruction, and noise resistance.