时变电磁问题的等几何多片预条件子

中文摘要 AI 辅助

我们基于高次样条几何方法的最新进展，针对麦克斯韦方程组的三维初边值问题展开研究。先前的方案仅限于单片几何。为了解决多片设置中质量矩阵求逆带来的计算挑战，我们将现有的0-形式质量矩阵和刚度矩阵技术扩展到1-形式质量矩阵。我们的方法利用样条空间的张量积结构，为1-形式质量矩阵构造一个谱等价的预条件子。具体而言，我们为单片域设计了一个高效的预条件子，该预条件子对网格尺寸和样条次数均具有鲁棒性。在多片情况下，通过将单片预条件子与加性Schwarz方法相结合来扩展该方法，确保了对片数的鲁棒性和可扩展性。我们分析了所提出预条件子的计算复杂度，并通过全面的数值实验验证了其鲁棒性和效率。

英文摘要

We build upon recent developments in high-order spline-based geometric methods for the three-dimensional initial boundary value problem of Maxwell's equations. Previous schemes were limited to single-patch geometries. To address the computational challenges associated with inverting mass matrices in multi-patch settings, we extend existing techniques for 0-forms mass and stiffness matrices to 1-forms mass matrices. Our approach leverages the tensor-product structure of spline spaces to construct a spectrally equivalent preconditioner for the 1-forms mass matrices. Specifically, we design an efficient preconditioner for single-patch domains that is robust with respect to both mesh size and spline degree. In the multi-patch case, this methodology is extended by integrating the single-patch preconditioner with an additive Schwarz approach, ensuring robustness and scalability with respect to the number of patches. We analyze the computational complexity of the proposed preconditioner and validate its robustness and efficiency through comprehensive numerical experiments.

发表机构

• École Polytechnique Fédérale Lausanne（洛桑联邦理工学院）
• Università di Pavia（帕维亚大学）
• Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” del CNR（意大利国家研究委员会应用数学与信息技术研究所（E. Magenes））
• Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学）
• Galician Centre for Mathematical Research and Technology（加利西亚数学研究与技术中心）

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