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Vertex-based Graph Neural Solver and its Application to Linear Elasticity Equations
arXiv:2511.21491v2 Announce Type: replace
Abstract: The efficient numerical solution of variable-coefficient linear elasticity equations on unstructured meshes presents a formidable challenge in computational mechanics. Convergence rates of classical iterative methods often stagnate due to material heterogeneity, strong anisotropy, and mesh irregularity. To address these limitations, this paper proposes a novel deep learning-based hybrid iterative method that integrates a weighted block Jacobi smoother with a graph neural network-enhanced global corrector. We introduce the adaptive graph Fourier neural solver, which employs a learnable coordinate transformation to construct a dynamic, problem-dependent spectral basis. This approach effectively overcomes the limitations of fixed Fourier bases in capturing the multi-scale features inherent in variable-coefficient media. Furthermore, to handle 3D and strongly anisotropic systems, we develop the multilevel adaptive graph Fourier neural solver, which executes hierarchical error correction across cascading frequency bandwidths. Rigorous theoretical analysis, grounded in the energy norm and Korn's inequality, establishes mesh-independent convergence guarantees. Extensive numerical experiments on 2D and 3D elasticity problems demonstrate that the proposed method exhibits superior robustness and convergence rates compared to the classical smoothed aggregation algebraic multigrid method, serving effectively as both a standalone solver and a preconditioner for Krylov subspace methods.
Abstract: The efficient numerical solution of variable-coefficient linear elasticity equations on unstructured meshes presents a formidable challenge in computational mechanics. Convergence rates of classical iterative methods often stagnate due to material heterogeneity, strong anisotropy, and mesh irregularity. To address these limitations, this paper proposes a novel deep learning-based hybrid iterative method that integrates a weighted block Jacobi smoother with a graph neural network-enhanced global corrector. We introduce the adaptive graph Fourier neural solver, which employs a learnable coordinate transformation to construct a dynamic, problem-dependent spectral basis. This approach effectively overcomes the limitations of fixed Fourier bases in capturing the multi-scale features inherent in variable-coefficient media. Furthermore, to handle 3D and strongly anisotropic systems, we develop the multilevel adaptive graph Fourier neural solver, which executes hierarchical error correction across cascading frequency bandwidths. Rigorous theoretical analysis, grounded in the energy norm and Korn's inequality, establishes mesh-independent convergence guarantees. Extensive numerical experiments on 2D and 3D elasticity problems demonstrate that the proposed method exhibits superior robustness and convergence rates compared to the classical smoothed aggregation algebraic multigrid method, serving effectively as both a standalone solver and a preconditioner for Krylov subspace methods.
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