= 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.
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