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

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.

Figures (6)

• Figure 1: Order of convergence for experiment 1.
• Figure 2: Snapshots of the magnetic spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) at given times in experiment 1. The colours indicate relative magnitude of the vectors.
• Figure 3: Order of convergence for experiment 2.
• Figure 4: Snapshots of the magnetic spin field $\boldsymbol{u}$ (projected onto $\mathbb{R}^2$) at given times in experiment 2. The colours indicate relative magnitude of the vectors.
• Figure 5: Order of convergence for experiment 3.

Theorems & Definitions (59)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 3.1
• proof