GMR-Net: GCN-based mesh refinement framework for elliptic PDE problems

Minseong Kim, Jaeseung Lee, Jibum Kim

Abstract: In this study, we propose a new approach for automatically generating high-quality non-uniform meshes based on supervised learning. The proposed framework, GMR-Net, is based on training a graph convolutional neural network (GCN) that learns local error density around each interior vertex in the mesh. The target edge length is then estimated using the predicted local error density. GMR-Net performs vertex-based error estimation as a preprocessing step to avoid expensive error estimation. The proposed GCN-based neural net has new input features for learning the local error density with various polygonal geometries, a range of partial differential equation parameters, and boundary conditions. During the test process, it predicts the target edge length around each interior vertex for the coarse input mesh, and the final non-uniform meshes are generated over the domain using Gmsh. To demonstrate the effectiveness of the GMR-Net, we tested the GMR-Net on Poisson’s equations and linear elasticity problems to previously unseen and new polygonal geometries and a range of PDE coefficients and boundary conditions.