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arXiv · 2304.10026

On a Sharp Estimate of Overlapping Schwarz Methods in $\mathrm{H}(\mathrm{curl};\Omega)$ and $\mathrm{H}(\mathrm{div};\Omega)$

Qigang Liang·Xuejun Xu·Shangyou Zhang
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Abstract

The previous proved-bound is $C(1+\frac{H^2}{\delta^2})$ for the condition number of the overlapping domain decomposition $\mathrm{H}(\mathrm{curl};\Omega)$ and $\mathrm{H}(\mathrm{div};\Omega)$ methods, where $H$ and $\delta$ are the sizes of subdomains and overlaps respectively. But all numerical results indicate that the best bound is $C(1+\frac{H}{\delta})$. In this work, we solve this long-standing open problem by proving that $C(1+\frac{H}{\delta})$ is indeed the best bound.

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Qigang Liang, Xuejun Xu, Shangyou Zhang. 2023-04-20. On a Sharp Estimate of Overlapping Schwarz Methods in $\mathrm{H}(\mathrm{curl};\Omega)$ and $\mathrm{H}(\mathrm{div};\Omega)$. https://arxiv.org/abs/2304.10026

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