  In this paper, we proposed two new types of edge multiscale methods motivated
by \cite{GL18} to solve Partial Differential Equations (PDEs) with
high-contrast heterogeneous coefficients: Edge spectral multiscale Finte
Element method (ESMsFEM) and Wavelet-based edge multiscale Finite Element
method (WEMsFEM). Their convergence rates for elliptic problems with
high-contrast heterogeneous coefficients are demonstrated in terms of the
coarse mesh size $H$, the number of spectral basis functions and the level of
the wavelet space $\ell$, which are verified by extensive numerical tests.
