Stable $C^1$-conforming finite element methods for a class of nonlinear fourth-order evolution equations
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913579419041792 |
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| author | Soenjaya, Agus L. Tran, Thanh |
| author_facet | Soenjaya, Agus L. Tran, Thanh |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_05530 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stable $C^1$-conforming finite element methods for a class of nonlinear fourth-order evolution equations Soenjaya, Agus L. Tran, Thanh Numerical Analysis 65M12, 65M60, 35Q60 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. |
| title | Stable $C^1$-conforming finite element methods for a class of nonlinear fourth-order evolution equations |
| topic | Numerical Analysis 65M12, 65M60, 35Q60 |
| url | https://arxiv.org/abs/2309.05530 |