基于力的准连续近似中的稳定性、不稳定性和误差
Stability, Instability, and Error of the Force-based Quasicontinuum Approximation
AI总结：
本文分析了基于力的准连续近似的稳定性，并利用稳定性结果获得了最优阶误差分析，证明了该方法的高精度。
摘要：
由于算法简单且精度高，基于力的模型耦合技术是计算物理学中一个令人兴奋的发展。例如，基于力的准连续近似是唯一已知的用于将一般原子模型与有限元连续模型耦合的点态一致准连续近似。本文分析了基于力的准连续近似的稳定性。然后，我们利用稳定性结果对该耦合方法进行了最优阶误差分析，为基于力的准连续近似的高精度提供了理论依据——如果缺陷在原子区域中被捕获，则可以在不损失显著精度的情况下利用连续建模的计算效率。我们需要克服的主要挑战是（我们证明了）线性化准连续算子通常不是正定的。此外，我们证明，对于离散版本的$W^{1,p}$-$W^{1,q}$“对偶配对”（其中$1/p+1/q=1$，且$1 \leq p < \infty$），不存在一致的inf-sup稳定性条件。因此，我们必须推导出离散版本$W^{1,\infty}$-$W^{1,1}$“对偶配对”的inf-sup稳定性条件，从而在离散$W^{1,\infty}$范数下得到最优阶误差估计。
英文摘要：
Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are an exciting development in computational physics. For example, the force-based quasicontinuum approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we analyze the stability of the force-based quasicontinuum approximation. We then use our stability result to obtain an optimal order error analysis of this coupling method that provides theoretical justification for the high accuracy of the force-based quasicontinuum approximation -- the computational efficiency of continuum modeling can be utilized without the loss of significant accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact (which we prove) that the linearized quasicontinuum operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the $W^{1,p}$-$W^{1,q}$ "duality pairing" with $1/p+1/q=1$, if $1 \leq p < \infty$. We must therefore derive an inf-sup stability condition for a discrete version of the $W^{1,\infty}$-$W^{1,1}$ "duality pairing" which then leads to optimal order error estimates in a discrete $W^{1,\infty}$-norm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
