Periodic Cartesian grids and tensor-product multilinear interpolation

Fix an integer $d\ge 1$ and integers $N_j\ge 3$, $1\le j\le d$. This is the unit-period specialization of the periodic tensor-grid definition in Periodic tensor grids and multilinear interpolation, with each period ${\mathrm{\ell }}_j=1$. Put $h_j=1\mathrm{/}{N}_j$ and define the periodic unit torus as ${\mathbb{T}}^d={\mathbb{R}}^d\mathrm{/}{\mathbb{Z}}^d$; functions on it are identified with ${\mathbb{Z}}^d$-periodic functions on ${\mathbb{R}}^d$. The Cartesian grid has nodes $x_i=(i_1{h}_1,\dots ,i_d{h}_d)(\mathrm{m}\mathrm{o}\mathrm{d}{\mathbb{Z}}^d),i_j\in \{0,\dots ,N_j-1\}\mathrm{.}$ Indices on nodal arrays are always reduced modulo $N_j$ in coordinate $j$.

For a nodal array $V=(V_i)$ and $x$ in the cell with lower corner $x_i$, write ${\tilde{x}}_j=(i_j+{\theta }_j)h_j$ in a periodic lift, where $0\le {\theta }_j\le 1$. Write $I_h$ for the supplier's tensor-product multilinear interpolant. In the local coordinates above, it is $I_{h}V(x)={\sum }_{ϵ\in \{0,1{\}}^d}({\prod }_{j=1}^d{\theta }_j^{{ϵ}_j}(1-{\theta }_j{)}^{1-{ϵ}_j})V_{i+ϵ}\mathrm{.}$ At cell faces either adjacent cell formula gives the same value, because the weight of the opposite face is zero. The formula is periodic and therefore defines $I_{h}V$ on ${\mathbb{T}}^d$.

In dimension two the four nodal weights on one cell are shown below. The dot is the evaluation point; the labels are exactly the four factors in the formula. The diagram fixes the cell convention only; no later proof relies on it.

For the lower-left node $V_{00}$ the weight is $(1-{\theta }_1)(1-{\theta }_2)$; the other three weights are ${\theta }_1(1-{\theta }_2)$, $(1-{\theta }_1){\theta }_2$, and ${\theta }_1{\theta }_2$. In every dimension the weights are nonnegative and sum to one by Periodic multilinear interpolation is convex and L-infinity contractive. The next local lemma proves the smooth interpolation error used by the schemes.