MeshFEM: A Block-accelerated Solver for Nonlinear Finite Elements

AlgMortar: a fully algebraic multiscale mortar preconditioner

The solution of large-scale symmetric positive definite linear systems arising from discretizations of second-order elliptic equations is challenging, especially in applications with highly heterogeneous coefficients, such as flow in porous media, which can lead to severely ill-conditioned systems. In this context, multiscale methods have recently been used to accelerate Krylov subspace methods, owing to their favorable parallel scalability. In this work, we present AlgMortar, a fully algebraic realization of the Multiscale Mortar Mixed Finite Element Method (MMMFEM). The method uses only information extracted from the fine-grid system matrix, which facilitates its implementation in existing solvers. AlgMortar uses graph partitioning to define a domain decomposition directly from the matrix graph. On each subdomain, it builds local linear systems that mimic Dirichlet problems, and couples the resulting local solutions through an algebraic interface condition that recovers the weak flux-continuity mechanism of MMMFEM. We prove that the method is well posed when the fine-grid matrix is symmetric positive definite and has nonpositive off-diagonal entries, a structure commonly arising from discretizations of elliptic problems. Numerical experiments on fine-grid linear systems arising from finite-volume discretizations of Darcy flow problems show that, when used as a preconditioner for the conjugate gradient method, the proposed approach exhibits good scalability and is competitive with state-of-the-art algebraic multigrid methods for challenging heterogeneous, high-contrast test cases, including highly irregular corner-point grids.

GPU-accelerated implicit solver for diffusion problems on unstructured grids

Implicit solvers on unstructured grids are widely used for diffusion-type problems because of their robustness and geometric flexibility. However, their efficiency in large-scale simulations is often limited by coefficient assembly, sparse linear-system solution, and irregular memory access caused by indirect grid connectivity. In this work, a GPUaccelerated implicit solver for diffusion problems is developed within the CMP platform. Two acceleration schemes are implemented and compared. In the hybrid CPU-GPU scheme, the coefficient matrix is assembled on the CPU, while only the sparse linear solver is executed on the GPU. In the full-process GPU scheme, discretization, matrix assembly, and linear solution are all performed on the GPU. AMGX is used for sparse linear solving, and the face-based discretization and assembly kernels on unstructured grids are redesigned in CUDA. To examine memory-access behavior, three data layouts, namely SoA, LAOS1, and LAOS2, are evaluated for the main face-based kernels. Four benchmark cases, including two-dimensional and three-dimensional configurations with both regular and complex geometries, are used to assess accuracy, convergence, scalability, memory behavior, and data-layout effects. The GPU solutions agree well with the CPU solutions and reference results, and the residual histories remain consistent among the computing schemes. Compared with the hybrid CPU-GPU scheme, the full-process GPU scheme achieves higher acceleration by reducing CPU-side assembly and repeated host-device transfer. For cases with more than one million cells, the measured speedup of the hybrid scheme reaches approximately 70% of the theoretical upper bound estimated by Amdahl's law. The data-layout results show that layout optimization has a limited effect in two-dimensional cases with relatively good locality, but becomes more effective for three-dimensional unstructured grids. By packing frequently co-accessed and indirectly accessed variables into localized AoS-like structures, LAOS can improve spatial locality on irregular three-dimensional grids and reduces scattered memory-access overhead. Among the tested layouts, LAOS1 provides the most stable overall improvement, whereas the benefit of LAOS2 depends more strongly on the access pattern of each kernel.

GPU-accelerated finite-element method for the three-dimensional unstructured mesh atmospheric dynamic framework

Graph Neural Multilevel Preconditioners for Iterative Solvers

Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning. While classical methods such as algebraic multigrid (AMG) are highly scalable, their robustness can degrade on indefinite or nonsymmetric systems where heuristics originally developed for elliptic PDEs are less reliable. Recently, Graph Neural Networks (GNNs) have emerged as data-driven preconditioners; yet, the practical impact of imposing an AMG-style hierarchy remains underexplored for general sparse matrices. In this work, we propose a Graph Neural Multilevel Preconditioner (GMP) that adopts an AMG hierarchy as a structural prior and learns smoothing, restriction, and interpolation operators in a unified framework. Our method targets general sparse systems and is instantiated as a drop-in preconditioner for standard Krylov solvers. On a benchmark of over 800 sparse matrices, we compare against classical AMG, single-level ILUT, and state-of-the-art GNN preconditioners, and characterize the regimes where multilevel graph neural preconditioning improves convergence or, conversely, introduces overhead relative to strong single-level baselines. These results highlight both the promise and the limitations of enforcing AMG-style multilevel structure in learned preconditioners for large-scale scientific simulations.

Parareal Algorithm For Gradient Flow Arising From Orthogonality-Constrained Matrix Optimization Problems

A Fully Matrix-Free Three-Grid Preconditioner for the Time-Harmonic Maxwell Equations at Extreme Scale

Three-dimensional time-harmonic Maxwell simulations generate massive complex indefinite systems whose mesh coarsening is strictly limited by phase accuracy. Although matrix-free finite element kernels utilize GPU throughput efficiently, standard multilevel solvers are ultimately bottlenecked by the memory and communication costs of exact coarse-grid factorizations. We present a fully matrix-free, factorization-free three-grid preconditioner for curl-conforming N{\'e}delec discretizations with perfectly matched layers (PML) and optimally blended quadrature. The method employs an outer FGMRES to solve the unshifted fine-grid equation, while an intermediate-grid correction is computed by a fixed-work FGMRES preconditioned with a complex-shifted $2h$--$4h$ cycle. This strategically confines the complex shift to an auxiliary preconditioner, preserving the physical Maxwell operator. A local Fourier analysis derives the blended Maxwell branches and compatible edge transfers, identifying robust shift and Jacobi damping parameters. Validated against the analytical Maxwell Green tensor, our approach demonstrates extreme scalability: using a single solver configuration, both homogeneous and highly heterogeneous systems with approximately 10.89 billion complex edge unknowns are solved in 42.0--72.0 seconds on just 64 NVIDIA A100 GPUs.