galerkin finite element discretization

This work proposes TensorGalerkin, a unified framework for efficiently solving variational-structure partial differential equations (PDEs), performing constrained optimization, and learning physics-informed operators. Built upon Galerkin discretization, the method tensorizes element-wise operations and maps them at the Python level, enabling differentiable message passing over mesh graphs through GPU-compatible sparse matrix multiplications. The resulting end-to-end differentiable architecture achieves substantial improvements in both computational efficiency and solution accuracy. Comprehensive benchmarks on 2D and 3D elliptic, parabolic, and hyperbolic PDEs demonstrate that TensorGalerkin consistently outperforms existing state-of-the-art baselines.

To address the computational inefficiency and trade-off between accuracy and efficiency in traditional adaptive finite element method (FEM) mesh generation for solving partial differential equations (PDEs) on complex 3D geometries—such as turbine blade scans—this paper proposes an end-to-end learning-based adaptive meshing framework. Our method employs a lightweight neural network to directly regress a spatial size field from sparse Monte Carlo (MC) solution estimates, replacing iterative optimization with a single forward inference pass. The pipeline integrates Monte Carlo sampling, neural regression, size-field-driven tetrahedral mesh generation, and FEM solving. Evaluated across diverse 3D shapes and boundary conditions, our approach achieves 2–4× speedup over conventional adaptive FEM and MC-based methods while maintaining comparable solution accuracy and demonstrating strong generalization capability.

This work addresses the high memory and computational complexity typically associated with solving three-dimensional partial differential equations on Cartesian grids. By exploiting tensor-product structure, the proposed method decomposes the 3D operator into one-dimensional banded kernels aligned with coordinate axes, thereby avoiding explicit assembly of the global matrix and enabling a matrix-free solution strategy. Within a unified framework that integrates diverse numerical approaches—including Kronecker product algebra, compact finite differences, isogeometric analysis, and direct diagonalization—the study systematically identifies three key techniques: multi-right-hand-side reshaping, sum factorization, and pencil-style MPI decomposition. These innovations collectively enhance hardware affinity and parallel scalability, reducing algorithmic complexity to O(N) and storage requirements to O(Nₓ + Nᵧ + N_z), thus enabling efficient large-scale 3D PDE simulations.

本文提出一种结合Gauss-Newton和Petrov-Galerkin方法的框架，用于解决偏微分方程数值解问题，通过线性测量离散化并选择测试函数来优化神经网络和混合有限元求解器。

This work addresses the widespread lack of built-in support for physical dimensions in existing finite element frameworks, which often leads to unit inconsistencies and numerical instabilities. For the first time, automated dimensional analysis is integrated into the Unified Form Language (UFL) by introducing a symbolic Quantity class that tracks physical units within variational forms. Leveraging the Abelian group structure of dimensions, units are encoded as rational-number vectors, and consistency checks along with nondimensionalization are automatically performed via a visitor pattern over expression trees. This approach reformulates nondimensionalization as a physics-aware diagonal preconditioner, substantially improving the condition number of saddle-point systems arising from the Navier–Stokes equations, identifying floating-point cancellation errors in Neo-Hookean hyperelastic models, and effectively handling parameter scaling in multiphysics Poisson–Nernst–Planck systems.

This study addresses the lack of systematic evaluation of hyper-reduction methods in nonlinear finite element models, particularly regarding the trade-off between accuracy and computational speedup. Within a unified open-source framework built on libROM, Laghos, and MFEM, the authors present the first reproducible benchmark comparing gappy POD interpolation and empirical quadrature (EQP) across diverse problems—nonlinear diffusion, elasticity, and Lagrangian hydrodynamics—and multiple time integrators. The results demonstrate that EQP achieves lower errors and higher efficiency with fewer integration points for diffusion and elasticity problems. However, in Lagrangian hydrodynamics, EQP incurs higher online costs, while the performance of interpolation-based methods is highly sensitive to the choice of time integration scheme. This work systematically reveals the dependence of hyper-reduction efficacy on both problem physics and numerical discretization choices.

本文提出了一种在黎曼流形上解决不可压缩Navier-Stokes方程的内在有限元方法，通过与表面FEM比较验证了其在长时间流动模拟中的准确性、效率及几何透明性。

This work addresses the challenge of generalizing partial differential equation (PDE) solvers to unseen geometric domains by proposing Geo-NeW, a novel method that jointly learns differential operators and compatible reduced finite element spaces within the framework of finite element exterior calculus to rigorously preserve physical conservation laws. Geo-NeW introduces geometry-aware neural Whitney forms that embed mesh geometric information into both Transformer encodings and basis function construction, thereby endowing neural PDE solvers with strong structure-preserving inductive biases. Furthermore, it devises a new constitutive model parameterization that guarantees the existence and uniqueness of solutions. Evaluated on multiple steady-state PDE benchmarks, the method achieves state-of-the-art performance and significantly outperforms conventional approaches on out-of-distribution geometries.

This study addresses the loss of precision caused by rounding errors in finite element computations, which remains difficult to analyze a priori. We propose the first automated a posteriori rounding error estimation framework that leverages running error analysis to track numerical and error propagation in real time. Built upon the FEniCS Form Compiler, this method achieves the first automated error estimation within finite element kernels through a C++ backend, custom arithmetic types, and templated kernel generation techniques. Experimental results demonstrate that the framework successfully detects catastrophic cancellation with only a 2–4× performance overhead. By effectively supporting mixed-precision design and numerical debugging, this work establishes a new paradigm for ensuring reliability in scientific computing.

This work addresses the limitations of traditional numerical methods—prohibitive computational cost in high-dimensional and geometrically complex settings—and the lack of theoretical guarantees and poor generalization in current machine learning approaches for solving partial differential equations (PDEs). To bridge this gap, the authors propose a hybrid PDE-solving paradigm that integrates deductive numerical schemes with inductive learning models. They establish a unified evaluation framework encompassing six core computational challenges and, from an epistemological perspective, formally distinguish between the two methodological classes. By introducing a structure inheritance mechanism and an error budget decomposition, they clarify the conditions under which theoretical guarantees propagate through the hybrid system. Leveraging physics-informed neural networks, differentiable programming, foundation models, and quantum algorithms, the study constructs a multi-paradigm collaborative framework and articulates responsible criteria for method selection, revealing three forms of complementarity that enable scalable, theoretically grounded simulation of high-dimensional complex systems.