Shifting the Sweet Spot: High-Performance Matrix-Free Method for High-Order Elasticity

Abstract: In high-order finite element analysis for elasticity, matrix-free (PA) methods are a key technology for overcoming the memory bottleneck of traditional Full Assembly (FA). However, existing implementations fail to fully exploit the special structure of modern CPU architectures and tensor-product elements, causing their performance "sweet spot" to anomalously remain at the low order of $p \approx 2$, which severely limits the potential of high-order methods. To address this challenge, we design and implement a highly optimized PA operator within the MFEM framework, deeply integrated with a Geometric Multigrid (GMG) preconditioner. Our multi-level optimization strategy includes replacing the original $O(p^6)$ generic algorithm with an efficient $O(p^4)$ one based on tensor factorization, exploiting Voigt symmetry to reduce redundant computations for the elasticity problem, and employing macro-kernel fusion to enhance data locality and break the memory bandwidth bottleneck. Extensive experiments on mainstream x86 and ARM architectures demonstrate that our method successfully shifts the performance "sweet spot" to the higher-order region of $p \ge 6$. Compared to the MFEM baseline, the optimized core operator (kernel) achieves speedups of 7x to 83x, which translates to a 3.6x to 16.8x end-to-end performance improvement in the complete solution process. This paper provides a validated and efficient practical path for conducting large-scale, high-order elasticity simulations on mainstream CPU hardware.