The discrete L2 norm

math

If a vector uh\mathbf{u}^h with components uihu_i^h is associated with a dd-dimensional domain with uniform grid spacing hh, then its discrete L2L^2 norm is given by

uhh=(hdi(uih)2)1/2,||\mathbf{u}^h||_{h} = \left(h^{d}\sum_{i} (u_i^h)^2 \right)^{1/2},

which is the usual Euclidean vector norm but scaled by a factor of hd/2h^{d/2}. The scaling factor makes the discrete L2L^2 norm a consistent approximation to the continuous L2L^2 norm of a function u(x)u(\mathbf{x}), which is given by

u2=(Ωu(x)2dx)1/2.||u||_{2} = \left( \int_{\Omega} |u(\mathbf{x})|^2 \, d\mathbf{x} \right)^{1/2}.

References

  • Briggs et al., A Multigrid Tutorial, pg 55

© 2024 Peter Cheng