A graph convolutional network (GCN) is employed in the deep energy method (DEM) model to solve the momentum balance equation in three‐dimensional space for the deformation of linear elastic and hyperelastic materials due to its ability to handle irregular domains over the traditional DEM method based on a multilayer perceptron (MLP) network. The method's accuracy and solution time are compared to the DEM model based on a MLP network. We demonstrate that the GCN‐based model delivers similar accuracy while having a shorter run time through numerical examples. Two different spatial gradient computation techniques, one based on automatic differentiation (AD) and the other based on shape function (SF) gradients, are also accessed. We provide a simple example to demonstrate the strain localization instability associated with the AD‐based gradient computation and show that the instability exists in more general cases by four numerical examples. The SF‐based gradient computation is shown to be more robust and delivers an accurate solution even at severe deformations. Therefore, the combination of the GCN‐based DEM model and SF‐based gradient computation is potentially a promising candidate for solving problems involving severe material and geometric nonlinearities.