We propose a novel efficient and robust Wavelet-based Edge Multiscale Finite Element Method (WEMsFEM) motivated by \cite{MR3980476,GL18} to solve the singularly perturbed convection-diffusion equations. The main idea is to first establish a local splitting of the solution over a local region by a local bubble part and local Harmonic extension part, and then derive a global splitting by means of Partition of Unity. This facilitates a representation of the solution as a summation of a global bubble part and a global Harmonic extension part, where the first part can be computed locally in parallel. To approximate the second part, we construct an edge multiscale ansatz space locally with hierarchical bases as the local boundary data that has a guaranteed approximation rate \noteLg{both inside and outside of the layers}. The key innovation of this proposed WEMsFEM lies in a provable convergence rate with little restriction on the mesh size. Its convergence rate with respect to the computational degree of freedom is rigorously analyzed, which is verified by extensive 2-d and 3-d numerical tests.

翻译：受\cite{MR3980476,GL18}启发，本文提出一种新颖高效且鲁棒的基于小波的边缘多尺度有限元方法（WEMsFEM）来求解奇异扰动对流扩散方程。其主要思想是：首先通过局部气泡函数部分和局部调和延拓部分在局部区域建立解的局部分裂，继而借助单位分解原理导出全局分裂。这使得解可表示为全局气泡部分与全局调和延拓部分之和，其中第一部分可在局部并行计算。为逼近第二部分，我们在局部构造具有分层基函数的边缘多尺度试探空间作为局部边界数据，该空间在边界层内部和外部均具有可保证的逼近速率。所提出的WEMsFEM的关键创新在于：在网格尺寸限制极小的条件下仍具有可证明的收敛速率。本文严格分析了该方法关于计算自由度的收敛率，并通过大量二维与三维数值实验验证了理论结果。