Discrete Sobolev norm ($\Vert W{\Vert }_{1,h}^2=h{\sum }_{i=1}^{N-1}|W_i{|}^2+h{\sum }_{i=1}^N|D_h^{-}{W}_i{|}^2$)

A discrete Sobolev norm combines weighted norms of nodal values and their finite differences. For a real sequence $W_0,\dots ,W_N$ with zero endpoints, the displayed expression is a norm: it is the Euclidean norm of one vector containing both weighted lists, and zero norm forces all nodal values to vanish. It approximates the Sobolev space $H^1$ norm. Endpoint weights or prescribed zero endpoints must be specified when calling a discrete expression a norm.