Computational Methods for Solving Partial Differential Equations: A Comparative Study of Finite-Element, Spectral and Meshless Schemes for Physics & Engineering Problems

Abstract

The paper provides a brief but strict benchmarking of three popular solvers of PDEs, namely FE, spectral and meshless schemes, with the current physics and engineering test-cases. New algorithmic advances to residual-driven hp-adaptivity, Trefftz DG, GPU-accelerated spectral elements, interpolation-based RKPM, and differentiable SPH have been mentioned as updates to the study to measure accuracy, CPU/GPU time, memory footprint and parallel scalability. Poisson, advection-diffusion and Navier-Stokes experiments have shown: (i) spectral methods retain exponential convergence when applied to smooth fields, but lose to discontinuities unless neural-network coordinate-changes; (ii) high-order FEM and HDG have the best balance between geometric flexibility, stability, and software maturity; (iii) meshless methods have been demonstrated to have the best performance in extreme deformation and free-surface flows, but require neighbour-search acceleration to match the speed of FEM. Moreover, graph-adaptive mesh refinement, spectral neural networks, physics-informed preconditioners, represent hybrid ML pipelines which reduce the number of iterations by up to 25 percent without affecting fidelity. An informaticized form of decision tree derived out of the data is used to direct practitioners in selecting what method would best work, whereas open-source scripts allow reproducibility and future extrapolation to exascale and quantum-augmented computing.