Stable $C^1$-conforming finite element methods for a class of nonlinear fourth-order evolution equations

TL;DR

This paper introduces stable, high-accuracy finite element methods for solving complex nonlinear fourth-order PDEs, including models like the Landau--Lifshitz--Baryakhtar and Cahn--Hilliard equations, with proven stability and convergence.

Contribution

It develops novel $C^1$-conforming finite element schemes for nonlinear fourth-order PDEs, providing stability analysis and optimal error estimates.

Findings

Schemes are stable in $\\mathbb{H}^2$ norm.

Achieve optimal convergence rates.

Numerical experiments confirm theoretical results.

Abstract

We propose some finite element schemes to solve a class of fourth-order nonlinear PDEs, which include the vector-valued Landau--Lifshitz--Baryakhtar equation, the Swift--Hohenberg equation, and various Cahn--Hilliard-type equations with source and convection terms, among others. The proposed numerical methods include a spatially semi-discrete scheme and two linearised fully-discrete $C^1$-conforming schemes utilising a semi-implicit Euler method and a semi-implicit BDF method. We show that these numerical schemes are stable in $\mathbb{H}^2$. Error analysis is performed which shows optimal convergence rates in each scheme. Numerical experiments corroborate our theoretical results.