A Robust Solver for a Mixed Finite Element Method for the Cahn-Hilliard Equation

TL;DR

This paper introduces a robust solver for a mixed finite element scheme solving the Cahn-Hilliard equation, featuring a preconditioned residual algorithm with multigrid preconditioning that performs efficiently regardless of mesh or time step sizes.

Contribution

It presents a novel preconditioned minimal residual solver with multigrid preconditioning that is mesh-independent and robust for the Cahn-Hilliard equation.

Findings

Solver performance is independent of mesh size.

Performance is also independent of time step size.

Dependence on interfacial width parameter is mild.

Abstract

We develop a robust solver for a mixed finite element convex splitting scheme for the Cahn-Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild.