Uniform Equivalence Of $L^2$ - and Discrete $ℓ^2$ -Norms on $Q_1$-Finite Element Spaces With Mass Lumping in Any Finite Dimension

The $Q_1$-finite element spaces, in any finite $d$-dimension, are equipped with the discrete $ℓ^2_h$-inner product generated by the simple row-sum mass lumping. The equivalence of the discrete $ℓ^2_h$-norm and the $L^2$-norm on the $Q_1$-finite element spaces is uniform in mesh size $h,$ in both cases of uniform and nonuniform partitions. Several representation formulae for these norms are derived. Using these, accurate bounds between these two norms are obtained, which is our major contribution. Examples show that these bounds are sharp. As an important application, the equivalence is established between discrete $h^1_h$-norm and $H^1$-norm. Numerical results are presented.