Empirical Analysis of Numerical Methods in Real-World PDE Problems

Thesis Abstract

The accurate solution of partial differential equations (PDEs) arising in engineering and physics hinges on the reliability and efficiency of numerical methods under real-world conditions. This study addresses the gap between theoretical convergence guarantees of numerical schemes and their empirical performance on complex, data-driven PDE models encountered in industry and research settings. The aim is to evaluate, benchmark, and quantify the accuracy, stability, and computational efficiency of prominent numerical methods across a representative suite of real-world PDE problems, thereby providing actionable guidance for method selection and algorithmic refinement. Specific objectives include (i) constructing a comprehensive benchmark corpus of real-world PDE problems from fluid mechanics, heat transfer, and geophysical modelling, with heterogeneous data inputs and irregular geometries; (ii) implementing and comparing finite difference, finite element, and spectral element methods under identical discretization controls on standardized hardware; (iii) assessing sensitivity to mesh resolution, time-stepping schemes, and solver configurations through systematic factorial experiments; (iv) employing robust statistical analyses to link problem characteristics (e.g., nonlinearity, anisotropy, stiffness) to numerical performance metrics; and (v) developing practical guidelines and a decision-support framework for practitioners. The methodology combines a multi-site, empirical field study with controlled experiments. The population comprises real-world PDE problems sourced from collaborating industrial partners and public datasets, including computational fluid dynamics flows around complex geometries, heat conduction in composite materials, and subsurface flow in heterogeneous porous media. A stratified sampling approach yields a test set of 60 PDE instances, each with known analytical or high-fidelity reference solutions where possible. For each instance, three numerical frameworks—finite difference, continuous Galerkin finite element, and spectral element methods—are implemented with consistent baseline solvers and preconditioners. Data collection instruments include standardized problem descriptions, mesh quality metrics, solver logs capturing iteration counts, convergence behavior, and wall-clock times, as well as error estimates relative to reference solutions. Validity and reliability are ensured through cross-validation against high-accuracy reference solutions (where feasible) and replication across at least three independent hardware environments to account for platform effects. The analysis employs a mixed-methods approach quantitative analysis using regression models and analysis of variance (ANOVA) to quantify the impact of method choice, mesh density, time-stepping, and problem class on accuracy, stability indicators (such as CFL compliance and spectral radius), and computational cost; complemented by a sensitivity analysis framework (Sobol indices) to identify dominant factors. An empirical evaluation metric suite includes L2 and H1 norm errors, energy dissipation rate discrepancies, and total solver time per degree of freedom. Theoretical underpinnings are anchored in the Lax equivalence framework and modern stability theories for discretized PDEs, with results interpreted through the lens of the Discontinuous Galerkin and Isogeometric Analysis perspectives where applicable. Expected findings indicate that, on real-world problems with irregular geometries and heterogeneous coefficients, finite element methods with adaptive mesh refinement and energy-stable time integrators yield the best overall balance between accuracy and efficiency, while spectral element methods excel in smooth, high-contrast domains given appropriate polynomial orders. The study will identify regimes where simpler finite difference discretizations remain competitive due to solver robustness and lower overhead. The contribution to knowledge lies in a comprehensive, empirically validated comparative framework for PDE numerical methods in practical settings, a benchmark corpus with real-world instances, and a decision-support toolkit for method selection and configuration grounded in statistical evidence and sensitivity analyses. The main conclusion anticipates that pragmatic method selection should be problem-class driven, with explicit guidelines on mesh strategy, time integration, and solver configurations; recommendations include prioritizing adaptive FEM with robust preconditioning for heterogeneous media and leveraging high-order spectral elements for smooth, large-scale simulations. The study will propose an open-access benchmarking portal and publish a replicable experimental protocol to facilitate ongoing empirical assessments by the computational science community.