Date of Award

2026-05-01

Degree Name

Master of Science

Department

Computational Science

Advisor(s)

Natasha S. Sharma

Abstract

Phase field models provide a versatile framework for describing phase transitions and pattern formation in materials, enabling the study of complex phenomena such as crystallization, grain growth, and defect dynamics. Among these models, the Phase Field Crystal (PFC) equation has gained significant attention due to its ability to capture atomic-scale structures while evolving on diffusive time scales. However, the numerical solution of the PFC equation presents substantial challenges arising from its high-order nature and the large-scale, coupled linear systems generated by discretization.

In particular, standard formulations of the discretized PFC system lead to non-symmetric and often ill-conditioned linear systems, which limit the use of efficient iterative solvers designed for symmetric systems and complicate the design of effective preconditioners. As a result, commonly used methods such as the Generalized Minimal Residual (GMRES) method may exhibit slow convergence and increased computational cost.

This thesis addresses these challenges by developing an efficient solver framework based on a fully discrete formulation for the PFC equation proposed by Backofen et al. in 2007. The formulation is rewritten to recover a symmetric structure, thereby enabling the use of the Minimal Residual (MINRES) method. In addition, a structured block preconditioner is constructed to improve the conditioning of the system and enhance the convergence behavior of the iterative solver. It is observed that applying the MINRES method to our preconditioned system we attain more than computational savings and as our mesh scales the computational speed up rises to a factor approximately 3 in our simulations.

Eigenvalue bounds for the preconditioned system are derived to characterize its spectral properties and to provide insight into the convergence behavior of the iterative solver. Numerical experiments demonstrate that the proposed approach yields improved convergence rates and computational efficiency across a range of discretization parameters.

Language

en

Provenance

Received from ProQuest

File Size

66 p.

File Format

application/pdf

Rights Holder

Raymond Obeng

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