Recognition of low-quality face images remains a challenge due to invisible or deformation in partial facial regions. For low-quality images dominated by missing partial facial regions, local region similarity contributes more to face recognition (FR). Conversely, in cases dominated by local face deformation, excessive attention to local regions may lead to misjudgments, while global features exhibit better robustness. However, most of the existing FR methods neglect the bias in feature quality of low-quality images introduced by different factors. To address this issue, we propose a Local and Global Feature Attention Fusion (LGAF) network based on feature quality. The network adaptively allocates attention between local and global features according to feature quality and obtains more discriminative and high-quality face features through local and global information complementarity. In addition, to effectively obtain fine-grained information at various scales and increase the separability of facial features in high-dimensional space, we introduce a Multi-Head Multi-Scale Local Feature Extraction (MHMS) module. Experimental results demonstrate that the LGAF achieves the best average performance on $4$ validation sets (CFP-FP, CPLFW, AgeDB, and CALFW), and the performance on TinyFace and SCFace outperforms the state-of-the-art methods (SoTA).