论文标题
多尺度线性椭圆PDE的指数收敛通过自适应边缘基础函数
Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions
论文作者
论文摘要
在本文中，我们引入了一个基于自适应边缘基础函数的多尺度框架，以求解具有粗糙系数的二阶线性椭圆PDES。主要结果之一是，我们证明所提出的多尺度方法在相对于自由度的近似误差中获得了几乎指数的收敛。我们的策略是将解决方案空间的能量正交分解为粗略的分量，该分量包括$ a $ harmonic函数在网格的每个元素中，以及一个名为气泡部分的精细比例分量，可以在本地和高效地计算。粗尺度组件完全取决于边缘上的函数值。我们在每个边缘上的近似值都是在狮子 - 马力空间中进行的，$ h_ {00}^{1/2}（e）$，我们将证明这是自然而有力的选择。我们使用局部过采样和单数值分解构建边缘基础函数。当将右侧的本地信息自适应地纳入边缘基础函数中时，我们证明近似误差的指数收敛速率几乎是指数的收敛速率。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的降解。
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 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.
