Wavelet-based Edge Multiscale Finite Element Methods for Singularly Perturbed Convection-Diffusion Equations

TL;DR

This paper introduces a wavelet-based edge multiscale finite element method for efficiently solving singularly perturbed convection-diffusion equations, with proven convergence and verified numerical results in 2D and 3D.

Contribution

It develops a novel wavelet-based multiscale finite element approach with a local solution splitting and hierarchical basis approximation, ensuring robustness and convergence for challenging equations.

Findings

Proven convergence rate with minimal mesh restrictions.

Effective approximation of solution layers inside and outside boundary layers.

Validated performance through extensive 2D and 3D numerical tests.

Abstract

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.