Development and Analysis of a Block-Preconditioner for the Phase-Field Crystal Equation

TL;DR

This paper introduces an efficient block-preconditioner for the linear systems from finite element discretizations of the Phase Field Crystal equation, enabling faster simulations of crystalline materials over long time scales.

Contribution

A novel block-preconditioner based on approximate factorization for the PFC equation's linear systems, improving computational efficiency and scalability.

Findings

Preconditioner significantly speeds up computations.

Enables scalable parallel algorithms for large systems.

Analyzed for effectiveness and efficiency.

Abstract

We develop a preconditioner for the linear system arising from a finite element discretization of the Phase Field Crystal (PFC) equation. The PFC model serves as an atomic description of crystalline materials on diffusive time scales and thus offers the opportunity to study long time behaviour of materials with atomic details. This requires adaptive time stepping and efficient time discretization schemes, for which we use an embedded Rosenbrock scheme. To resolve spatial scales of practical relevance, parallel algorithms are also required, which scale to large numbers of processors. The developed preconditioner provides such a tool. It is based on an approximate factorization of the system matrix and can be implemented efficiently. The preconditioner is analyzed in detail and shown to speed up the computation drastically.