Stable C¹-conforming finite element methods for a class of nonlinear fourth-order evolution equations

2023/09/11 by Agus L. Soenjaya, Thanh Tran, Soenjaya, Agus L. +1 · 1 citation
Computer Science · Mathematics · #math.NA #cs.NA

paper · pdf · doi:10.48550/arxiv.2309.05530

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 C¹-conforming schemes utilising a semi-implicit Euler method and a semi-implicit BDF method. We show that these numerical schemes are stable in ℍ². Error analysis is performed which shows optimal convergence rates in each scheme. Numerical experiments corroborate our theoretical results.

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• A mixed finite element method for a class of fourth-order stochastic evolution equations with multiplicative noise

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