Empirical Analysis of Numerical Methods for Large-Scale PDE Solvers

Thesis Abstract

This study addresses the practical performance and reliability of numerical methods used for solving large-scale partial differential equations (PDEs) in scientific computing, focusing on how method choice, discretization strategy, and solver configurations influence accuracy, convergence, and computational efficiency in real-world workflows. The aim is to empirically evaluate a representative set of numerical techniques—including finite element, finite volume, and spectral methods—across canonical large-scale PDE problems with heterogeneous coefficients and complex domain geometries to identify robust combinations of discretization and solver strategies. Specific objectives are (1) to quantify accuracy versus computational cost for iterative solvers (Krylov subspace methods, multigrid, and domain decomposition) applied to structured and unstructured meshes up to 10^7 degrees of freedom; (2) to assess the impact of preconditioning strategies (ILU, algebraic multigrid, and block preconditioners) on convergence rates under varying mesh resolutions and problem conditioning; (3) to compare time-to-solution and energy consumption on conventional CPU and heterogeneous GPU-accelerated architectures; (4) to analyze sensitivity to discretization parameters (mesh size, time step, and stabilization techniques) through global error metrics and local residual distributions; and (5) to develop practical guidelines for selecting numerical methods in large-scale simulations common to fluid dynamics, solid mechanics, and heat transfer. The methodology adopts a mixed-methods empirical design combining controlled computational experiments with statistical analysis. The population comprises three representative PDE classes—Stokes/Navier–Stokes for incompressible flow, Poisson/elliptic problems for diffusion, and advection–diffusion–reaction systems—solved on synthetic benchmark domains and a high-fidelity continental-scale groundwater flow domain provided by a public hydrogeological dataset. A stratified sampling approach constructs a factorial experiment with factors including discretization method (finite element, finite volume, spectral), mesh density (coarse, medium, fine, extra-fine), time-stepping scheme (implicit, explicit, semi-implicit), solver type (CG, GMRES, BiCGSTAB, and multigrid variants), and preconditioner configuration. The total experimental runs exceed 1,200, with each configuration executed on high-performance computing platforms at two sites to capture hardware variability. Data collection instruments include (i) precise runtime traces and scalability metrics captured via integrated profiling tools (TAU, Intel VTune, NVIDIA Nsight); (ii) numerical accuracy indicators such as L2 and H1 norm errors against analytical or high-fidelity reference solutions; (iii) iteration counts, convergence histories, and residual norms; (iv) memory footprints and cache/memory bandwidth utilization; (v) energy consumption data gathered through power meters and platform-specific energy APIs; and (vi) qualitative logs of solver stability flags and failure modes. Validity and reliability are ensured through duplicate runs, randomized initial conditions for stochastic solvers, cross-platform normalization of performance metrics, and calibration against known analytic benchmarks. Analytical methods include regression analysis to model the relationship between discretization choices and error versus time, ANOVA to detect significant main effects and interactions among factors, and nonparametric tests where normality assumptions fail. Descriptive statistics summarize stability and robustness across problem classes. A meta-modeling step develops an actionable decision-support framework that maps problem features (PDE type, coefficient heterogeneity, domain geometry) to recommended method configurations. Theoretical grounding draws on established spectral accuracy and stability theories, while the study also engages the Mertonian view of computational experiments as a means to generalize beyond specific hardware configurations. Expected findings indicate that (a) spectral methods offer superior accuracy per DOF for smooth problems but incur higher implementation complexity and memory overhead; (b) algebraic multigrid preconditioning consistently enhances convergence rates for elliptic and diffusion-dominated problems across scales, with diminishing returns on highly anisotropic meshes; (c) domain decomposition and GPU-accelerated solvers provide substantial speedups for highly parallel workloads, though communication overhead and preconditioner parallelism become critical bottlenecks; (d) time discretization choices significantly affect stability and energy consumption, with implicit schemes favored for stiff advection–diffusion problems and semi-implicit schemes balancing accuracy and cost in mixed regimes. The study contributes to knowledge by delivering a comprehensive, empirically validated assessment of large-scale PDE solver methods, producing a practical guideline framework for method selection under real-world computational constraints, and informing future solver development to enhance scalability, robustness, and energy efficiency. The main conclusion anticipates that integrated, hardware-aware solver pipelines combining robust preconditioned iterative methods with adaptive discretization strategies yield the best balance of accuracy, efficiency, and resilience for diverse large-scale PDE applications. Recommendations include adopting hybrid discretizations with problem-adapted preconditioners, prioritizing multigrid-accelerated solvers for elliptic components, and investing in energy-aware performance profiling to optimize large-scale simulations.