非线性Oldroyd方程带一般记忆核的增量SVD压缩
Incremental SVD Compression for Nonlinear Oldroyd Equations with General Memory Kernels
AI总结：
本文研究非线性Oldroyd问题的混合有限元/Crank-Nicolson离散化方法，通过增量SVD压缩历史项以降低存储和计算开销，适用于一般非奇异和弱奇异记忆核。
摘要：
我们研究了非线性Oldroyd问题带一般非奇异和弱奇异记忆核的混合有限元/Crank--Nicolson离散化。直接计算历史项需要存储所有先前速度快照，导致存储复杂度为$\mathcal{O}(mN)$，计算复杂度为$\mathcal{O}(mN^2)$。为减少此负担，我们通过增量奇异值分解压缩速度历史，并在离散记忆项中使用压缩表示。在数值秩为$r$的近似假设下，存储复杂度降低到$\mathcal{O}((m+N)r)$，总历史计算复杂度变为$\mathcal{O}(mNr+rN^2)$。对于非奇异核，我们推导出依赖于容差的扰动估计，表明当压缩容差足够小时，基有限元精度得以保持。我们还通过卷积四阶扩展该方法至温控弱奇异核。数值测试显示，在报告的容差下，解与未压缩方案几乎无法区分，同时实现显著的内存节省和计算时间减少。
英文摘要：
We study mixed finite element/Crank--Nicolson discretizations of a nonlinear Oldroyd problem with general nonsingular and weakly singular memory kernels. Direct evaluation of the history term requires storing all previous velocity snapshots, which leads to $\mathcal{O}(mN)$ memory and $\mathcal{O}(mN^2)$ work over $N$ time steps, where $m$ denotes the number of spatial degrees of freedom. To reduce this burden, we compress the velocity history online by an incremental singular value decomposition and use the compressed representation in the discrete memory term. Under an approximate low-rank assumption of numerical rank $r$, the storage decreases to $\mathcal{O}((m+N)r)$, while the total history-evaluation work becomes $\mathcal{O}(mNr+rN^2)$. For nonsingular kernels, we derive a tolerance-dependent perturbation estimate showing that the baseline finite element accuracy is retained when the compression tolerance is sufficiently small. We also extend the approach to tempered weakly singular kernels via convolution quadrature. Numerical tests show near-indistinguishable solutions from the uncompressed scheme for the reported tolerances, together with substantial memory savings and reduced wall-clock time.
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