Multiscale Modeling &Simulation, Volume 19, Issue 2, Page 980-1010, January 2021.
In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove that the proposed multiscale method achieves nearly exponential convergence in the approximation error with respect to the computational degrees of freedom. Our strategy is to perform an energy orthogonal decomposition of the solution space into a coarse scale component comprising $a$-harmonic functions in each element of the mesh, and a fine scale component named the bubble part that can be computed locally and efficiently. The coarse scale component depends entirely on function values on edges. Our approximation on each edge is made in the Lions--Magenes space $H_{00}^{1/2}(e)$, which we will demonstrate to be a natural and powerful choice. We construct edge basis functions using local oversampling and singular value decomposition. When local information of the right-hand side is adaptively incorporated into the edge basis functions, we prove a nearly exponential convergence rate of the approximation error. Numerical experiments validate and extend our theoretical analysis; in particular, we observe no obvious degradation in accuracy for high-contrast media problems.

中文翻译：

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通过自适应边基函数实现多尺度线性椭圆偏微分方程的指数收敛

多尺度建模与仿真，第 19 卷，第 2 期，第 980-1010 页，2021 年 1 月。
在本文中，我们引入了一种基于自适应边缘基函数的多尺度框架来求解具有粗糙系数的二阶线性椭圆偏微分方程。主要结果之一是我们证明了所提出的多尺度方法在相对于计算自由度的近似误差方面实现了近乎指数收敛。我们的策略是将解空间的能量正交分解为粗尺度分量，包括网格的每个元素中的$a$-谐波函数，以及可以局部有效计算的称为气泡部分的细尺度分量。粗尺度分量完全取决于边上的函数值。我们对每条边的近似是在 Lions--Magenes 空间 $H_{00}^{1/2}(e)$ 中进行的，我们将证明这是一个自然而强大的选择。我们使用局部过采样和奇异值分解构造边缘基函数。当右侧的局部信息自适应地合并到边缘基函数中时，我们证明了近似误差的近指数收敛速度。数值实验验证并扩展了我们的理论分析；特别是，我们观察到高对比度媒体问题的准确性没有明显下降。