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Error Bounds for Discrete Minimizers of the Ginzburg–Landau Energy in the High-[math] Regime
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2024-05-30 , DOI: 10.1137/23m1560938 Benjamin Dörich 1 , Patrick Henning 2
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2024-05-30 , DOI: 10.1137/23m1560938 Benjamin Dörich 1 , Patrick Henning 2
Affiliation
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1313-1343, June 2024.
Abstract. In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter [math]. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of [math] into a mesh resolution condition, which can be done through error estimates that are explicit with respect to [math] and the spatial mesh width [math]. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted [math]-weighted norm. Afterward we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived [math]- and [math]-error estimates are indeed optimal in [math] and [math].
中文翻译:
高[数学]体系中金兹堡-朗道能量离散极小化器的误差界
《SIAM 数值分析杂志》,第 62 卷,第 3 期,第 1313-1343 页,2024 年 6 月。
抽象的。在这项工作中,我们研究有限元空间中金兹堡-朗道能量的离散最小化器。特别关注 Ginzburg-Landau 参数 [数学] 的影响。该参数具有物理意义,因为较大的值可以触发涡旋晶格的出现。由于必须在足够精细的计算网格上解析涡流,因此将 [math] 的大小转换为网格分辨率条件非常重要,这可以通过相对于 [math] 和空间网格明确的误差估计来完成宽度[数学]。为此,我们首先在一般类离散空间的抽象框架中工作,在该框架中我们以适应问题的[数学]加权范数呈现收敛结果。然后,我们将我们的发现应用于拉格朗日有限元和特定的广义有限元构造。在数值实验中,我们确认我们导出的 [math]- 和 [math]- 误差估计在 [math] 和 [math] 中确实是最优的。
更新日期:2024-05-29
Abstract. In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter [math]. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of [math] into a mesh resolution condition, which can be done through error estimates that are explicit with respect to [math] and the spatial mesh width [math]. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted [math]-weighted norm. Afterward we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived [math]- and [math]-error estimates are indeed optimal in [math] and [math].
中文翻译:
高[数学]体系中金兹堡-朗道能量离散极小化器的误差界
《SIAM 数值分析杂志》,第 62 卷,第 3 期,第 1313-1343 页,2024 年 6 月。
抽象的。在这项工作中,我们研究有限元空间中金兹堡-朗道能量的离散最小化器。特别关注 Ginzburg-Landau 参数 [数学] 的影响。该参数具有物理意义,因为较大的值可以触发涡旋晶格的出现。由于必须在足够精细的计算网格上解析涡流,因此将 [math] 的大小转换为网格分辨率条件非常重要,这可以通过相对于 [math] 和空间网格明确的误差估计来完成宽度[数学]。为此,我们首先在一般类离散空间的抽象框架中工作,在该框架中我们以适应问题的[数学]加权范数呈现收敛结果。然后,我们将我们的发现应用于拉格朗日有限元和特定的广义有限元构造。在数值实验中,我们确认我们导出的 [math]- 和 [math]- 误差估计在 [math] 和 [math] 中确实是最优的。