L2-compatible initialization of grid data

An arbitrary $L^2$ initial function has no well-defined nodal point samples. Cell averages or a bounded projection to the discrete approximation space supply grid data independently of its pointwise representative. Jensen's inequality bounds the cell-average discrete L2 norm by the continuous L2 norm. Combining consistent bounded initialization, stability, and density of smooth data can extend convergence to rough initial functions, without claiming their nonexistent classical derivatives.

The cell-average projection replaces a function by its average on each grid cell. It is independent of the representative of an L2 function, unlike point sampling. The Cauchy-Schwarz inequality gives $h|U_m{|}^2\le {\int }_{\text{cell }m}|u{|}^{2}dx$, hence $h{\sum }_m|U_m{|}^2\le \Vert u{\Vert }_{L^2}^2$. Viewed as a piecewise constant function, the projection is the orthogonal projection onto the grid's piecewise constant linear subspace. Density of continuous compactly supported functions and the contraction estimate give convergence in the L2 norm as the mesh tends to zero.