A preconditioner for the Ohta--Kawasaki equation

TL;DR

This paper introduces a novel preconditioner for the nonlocal Ohta--Kawasaki equation, enabling efficient, mesh-independent, and robust large-scale simulations of diblock copolymer melts.

Contribution

A new preconditioner for the Ohta--Kawasaki equation that ensures mesh independence and robustness, facilitating large-scale 3D simulations.

Findings

Preconditioner achieves mesh-independent convergence.

Robustness with respect to interfacial thickness parameter.

Enables simulation of over one billion degrees of freedom.

Abstract

We propose a new preconditioner for the Ohta--Kawasaki equation, a nonlocal Cahn--Hilliard equation that describes the evolution of diblock copolymer melts. We devise a computable approximation to the inverse of the Schur complement of the coupled second-order formulation via a matching strategy. The preconditioner achieves mesh independence: as the mesh is refined, the number of Krylov iterations required for its solution remains approximately constant. In addition, the preconditioner is robust with respect to the interfacial thickness parameter if a timestep criterion is satisfied. This enables the highly resolved finite element simulation of three-dimensional diblock copolymer melts with over one billion degrees of freedom.