Deep learning-based approaches, particularly graph neural networks (GNNs), have gained prominence in simulating flexible deformations and contacts of solids, due to their ability to handle unstructured physical fields and nonlinear regression on graph structures. However, existing GNNs commonly represent meshes with graphs built solely from vertices and edges. These approaches tend to overlook higher-dimensional spatial features, e.g., 2D facets and 3D cells, from the original geometry. As a result, it is challenging to accurately capture boundary representations and volumetric characteristics, though this information is critically important for modeling contact interactions and internal physical quantity propagation, particularly under sparse mesh discretization. In this paper, we introduce MAVEN, a mesh-aware volumetric encoding network for simulating 3D flexible deformation, which explicitly models geometric mesh elements of higher dimension to achieve a more accurate and natural physical simulation. MAVEN establishes learnable mappings among 3D cells, 2D facets, and vertices, enabling flexible mutual transformations. Explicit geometric features are incorporated into the model to alleviate the burden of implicitly learning geometric patterns. Experimental results show that MAVEN consistently achieves state-of-the-art performance across established datasets and a novel metal stretch-bending task featuring large deformations and prolonged contacts.