Aspect-level sentiment analysis (ALSA) aims to identify the sentiment polarity associated with specific aspects in textual data. However, existing methods utilizing graph convolutional networks (GCNs) face significant challenges, particularly in analyzing sentiments for multi-word aspects and capturing sentiment relationships across multiple aspects in complex sentences. To address these issues, we introduce the Specific-aspect and Inter-aspect Graph Convolutional Network (SI-GCN), which integrates contextual information, syntactic dependencies, and commonsense knowledge to provide a robust solution. The SI-GCN model incorporates several innovative components: a Specific-aspect GCN module that effectively captures sentiment features for individual aspects; a knowledge-enhanced heterogeneous graph designed to manage implicit sentiment expressions and multi-word aspects; and a dual affine attention mechanism that accurately models inter-aspect relationships. Compared to existing state-of-the-art methods, the SI-GCN achieves improvements in performance ranging from 0.9% to 2.3% across four benchmark datasets. A detailed analysis of text semantics shows that the SI-GCN excels in challenging scenarios, including those involving aspects without explicit sentiment indicators, multi-word aspects, and informal language structures.