Impact of Nested Preconditioning in hp-Adaptive Continuous Galerkin Solutions of the Poisson Problem

Abstract

This thesis investigates the performance of several preconditioning strategies for the Conjugate Gradient (CG) method applied to finite element discretization of the Poisson equation in one and two dimensions. Both uniform and adaptive refinement approaches are considered, including 𝑝-refinement, ℎ-refinement, and the combined ℎ𝑝-adaptivity strategy. The study evaluates nine different preconditioners, including classical methods such as Jacobi and SSOR, low order finite element-based (order 1) preconditioners (FEM1, F3T, F4R, F4CR), 𝑝-multigrid techniques, and multi-preconditioned conjugate gradient (MPCG) schemes.

The analysis is carried out systematically by examining three aspects: iteration counts, execution times, and condition numbers, with particular attention given to eigenvalue distributions. Results show that condition number alone is not a sufficient predictor of solver performance. Instead, the clustering and spread of eigenvalues play a decisive role in determining the convergence rate of CG. Preconditioners such as FEM1 in one dimension, and F3T or F4R in two dimensions, demonstrate the ability to cluster eigenvalues into compact groups, which significantly reduce iteration counts compared to Jacobi and SSOR. In contrast, preconditioners like F4CR exhibit very small condition numbers but still require many more iterations due to poor eigenvalue clustering.

The uniform refinement results highlight the reliability of the 𝑝-multigrid method, whose iteration counts remain nearly independent of system size and polynomial degree. However, its runtime efficiency only becomes evident at large matrix sizes, due to the overhead of inter-level transfers at smaller scales. The adaptive refinement studies reveal distinct behaviors: 𝑝-adaptivity consistently produced moderate system sizes with stable iteration counts, ℎ-adaptivity led to prohibitively expensive runtimes despite moderate number of iterations, and ℎ𝑝-adaptivity achieved the best balance, requiring roughly the same number of iterations as 𝑝-adaptivity while generating much larger systems.

Overall, the results emphasize that the effectiveness of preconditioners must be judged not only by condition number reduction, but also by their impact on eigenvalue distribution and clustering, which ultimately governs CG performance.