Spectral ACMS：一种鲁棒的局部化近似分量模态综合方法
Spectral ACMS: A robust localized Approximated Component Mode Synthesis Method
AI总结：
针对含非均匀系数的二维对称椭圆偏微分方程，提出两种多尺度Galerkin有限元方法，第二种通过添加边模态实现与网格、对比度无关的最优误差估计，且对低正则性解仍最优收敛。
摘要：
我们研究用于逼近含非均匀$L^\infty$系数的二维对称椭圆偏微分方程解的多尺度型有限元方法。这类方法属于Galerkin型，遵循变分多尺度方法与局部正交分解（LOD）方法的思路，将空间解耦为多尺度子空间与精细子空间。第一种方法中，基于原始迭代子结构方法所用角点，将粗基函数映射为全局最小能量函数，从而得到多尺度基函数。该方法在网格尺寸方面可实现拟最优先验误差能量逼近，但对高对比度系数不具备鲁棒性。第二种方法将基于局部广义特征值问题的边模态添加到角模态中，最终得到与网格和对比度均无关的最优先验误差能量估计。即使解仅具有最低正则性（仅属于Sobolev空间$H^1$），这些方法仍能以最优速率收敛。
英文摘要：
We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into \emph{multiscale} and \emph{fine} subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space $H^1$.
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