Comparative Analysis of Spectral Methods for PDEs on Irregular Domains

Thesis Abstract

This study investigates the comparative performance of spectral methods for solving partial differential equations (PDEs) on irregular domains, addressing a gap where domain geometry undermines the accuracy and efficiency of classical spectral techniques. The problem stems from the high-accuracy potential of spectral methods being compromised by complex boundaries, leading to spurious oscillations, poor convergence, and increased computational cost. The aim is to systematically evaluate and contrast Fourier, Chebyshev, and Legendre spectral representations, along with hybrid domain-decomposition approaches, to determine their relative strengths, limitations, and applicability to irregular geometries. Specific objectives include (i) to quantify accuracy and convergence rates of each method against analytical and benchmark solutions on irregular domains; (ii) to assess stability, conditioning, and error propagation under varying mesh refinements and mesh irregularities; (iii) to examine computational efficiency and memory usage across problem sizes; (iv) to evaluate the impact of interface treatments in hybrid methods on solution quality; and (v) to develop practical guidelines for method selection tailored to PDE type, boundary conditions, and domain geometry. The methodology adopts a comparative, quantitative research design with a mixed-methods flavor to triangulate numerical performance metrics and theoretical insights. The population comprises canonical PDEs representative of elliptic, parabolic, and hyperbolic classes, including Poisson, heat, and wave equations posed on irregular domains in two and three dimensions. A stratified sample of test problems is constructed with domain geometries that introduce curvature, re-entrant corners, and non-convex features. For each PDE, numerical experiments are conducted using (a) pure spectral methods with global basis functions (Fourier for periodic mappings, Chebyshev and Legendre for nonperiodic domains), (b) domain-decomposition spectral methods with interface continuity constraints, and (c) hybrid spectral–finite element approaches leveraging mortar or penalty methods. Data collection instruments comprise high-precision reference solvers (finite element and finite difference schemes) and standardized error norms (L2 and L?) computed against analytical solutions where available or highly resolved benchmark solutions. Additional metrics include condition numbers of discretized operators, iteration counts for iterative solvers, and wall-clock times on multi-resolution grids. Analytical techniques deployed encompass regression analysis to model convergence behavior as a function of mesh density and domain irregularity, ANOVA to compare performance across method categories, and multi-criteria decision analysis to weigh accuracy against computational cost. The study also employs spectral gap analysis and Fourier/Legendre coefficient decay profiling to interpret convergence properties and aliasing effects. Theoretical underpinnings draw on approximation theory, insights from Kolmogorov n-widths, and the Abstract
Approximation Theorem, with explicit reference to the Galerkin framework and stability proofs for non-periodic spectral bases. The expected findings indicate that domain irregularities degrade uniform convergence for global spectral bases, with Chebyshev and Legendre collocation methods exhibiting superior accuracy near boundaries, while domain-decomposition and hybrid methods mitigate spurious boundary effects and restore near-optimal convergence rates. It is anticipated that Fourier-based methods under irregular mappings may suffer from aliasing unless carefully treated with filtering or transformed coordinates. Computational efficiency is projected to favor domain-decomposition strategies for fine-scale irregular domains, whereas pure spectral methods may outperform others on smoother interfaces with moderate sizes. The study contributes to knowledge by providing a rigorous, empirical comparison of spectral methods on irregular domains, offering a decision framework and practical guidelines for method selection, error control strategies, and implementation choices in scientific computing environments. It also advances understanding of how interface treatment and hybridization influence stability and accuracy, with implications for high-performance computing applications in physics, engineering, and applied mathematics. The main conclusion is expected to emphasize that no single spectral approach universally outperforms others; instead, the choice should be governed by domain geometry, desired accuracy, and available computational resources. Recommendations include adopting hybrid spectral–finite element schemes for highly irregular geometries, applying spectral filtering to control aliasing in global bases, and developing adaptive domain-decomposition strategies that concentrate spectral resolution where boundary features are most pronounced.