The Landau--Lifshitz--Bloch equation with spin diffusion: Global strong solution and finite element approximation
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| Format: | Preprint |
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2026
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| _version_ | 1866918425361645568 |
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| author | Soenjaya, Agus L. |
| author_facet | Soenjaya, Agus L. |
| contents | The spin-diffusion Landau--Lifshitz--Bloch (SDLLB) system is a nonlinearly coupled system of quasilinear vector-valued PDEs which models the interaction between spin-polarised currents and magnetisation at high temperatures. The aim of this paper is twofold. Firstly, assuming the initial data is sufficiently small, we show the existence of a unique global strong solution to the SDLLB equation in a bounded domain $Ω\subset \mathbb{R}^d$, where $d\leq 3$, thus ensuring well-posedness of the model. Secondly, we propose a decoupled linearised fully-discrete finite element scheme to solve the problem. Despite the strong nonlinearity of the system, the proposed scheme only requires the solution of two completely decoupled linear systems per time-step. Assuming adequate regularity of the exact solution and a certain time-step constraint, we rigorously show that the numerical scheme converges at an optimal rate. Several numerical experiments corroborate our theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01668 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Landau--Lifshitz--Bloch equation with spin diffusion: Global strong solution and finite element approximation Soenjaya, Agus L. Numerical Analysis Analysis of PDEs 65M12, 65M60, 35K59, 35Q60 The spin-diffusion Landau--Lifshitz--Bloch (SDLLB) system is a nonlinearly coupled system of quasilinear vector-valued PDEs which models the interaction between spin-polarised currents and magnetisation at high temperatures. The aim of this paper is twofold. Firstly, assuming the initial data is sufficiently small, we show the existence of a unique global strong solution to the SDLLB equation in a bounded domain $Ω\subset \mathbb{R}^d$, where $d\leq 3$, thus ensuring well-posedness of the model. Secondly, we propose a decoupled linearised fully-discrete finite element scheme to solve the problem. Despite the strong nonlinearity of the system, the proposed scheme only requires the solution of two completely decoupled linear systems per time-step. Assuming adequate regularity of the exact solution and a certain time-step constraint, we rigorously show that the numerical scheme converges at an optimal rate. Several numerical experiments corroborate our theoretical results. |
| title | The Landau--Lifshitz--Bloch equation with spin diffusion: Global strong solution and finite element approximation |
| topic | Numerical Analysis Analysis of PDEs 65M12, 65M60, 35K59, 35Q60 |
| url | https://arxiv.org/abs/2604.01668 |