Community discovery is an important task for graph mining. Owing to the nonstructure, the high dimensionality, and the sparsity of graph data, it is not easy to obtain an appropriate community partition. In this paper, a deep graph convolution neural network with non-negative matrix factorization (DGCN-NMF) is proposed for community discovery. DGCN-NMF employs multiple graph convolution layers to obtain the low dimensional embedding. In each layer, the last output and the non-negative matrix factorization's results of the previous outputs are inputted into the current layer. The community partition of the inputted graph can be outputted by an end-to-end approach. The proposed algorithm and the comparison algorithms are conducted on the experimental data sets. The experimental results show that the proposed algorithm outperforms the comparison algorithms on the experimental data sets. The experimental results demonstrate that DGCN-NMF is an effective algorithm for community discovery.