Some notation before introducing the question. We discretize an interval $[0,L]$ in uniform subintervals, each with length $\Delta x$ and we assume that vectors living over such a grid satisfies periodic boundary conditions, that is to say $u_0 = u_{n}$.
We introduce the following scalar product: $ (v,w) := \sum_0^{n-1} v_i w_i \Delta x$, and we denote by $|| \cdot ||$ the norm that such a product induces.
We denote by $ w * v$ the Hadamard product, or component-wise product: $ [w * v]_{i} = w_{i}v_{i} $. It holds in general that $ || v*w || \leq \frac{1}{\sqrt{\Delta x}} ||w || || v||$.
We make use of the following matrices: - $ D $, matrix coming from the centred first order discretization with periodic boundary conditions of the first derivative using finite differences.\ - $ Q $ discrete Laplace operator with periodic boundary conditions.\ - $I$ is an identity matrix.\ Everything is assumed to have the right dimension, so that what comes next makes sense
I need to estimate the following expression: $$ || (I - Q)^{-1}\Big( (I-Q)u * Dv \Big) ||.$$ The naive way to do that would be $$ || (I - Q)^{-1}\Big( (I-Q)u * Dv \Big) || \leq || (I - Q)^{-1}|| \frac{1}{\sqrt{\Delta x}} \ ||(I-Q)|| \ || u|| \ || D || || v || $$ which would give, up to a constant, a bound of the form $ \frac{1}{\Delta x ^{\frac{7}{2}}} $.
This because $ || D || \leq \frac{1}{\Delta x}$, $||(I-Q)^{-1} || \leq 1$ and $ || (Q) || \leq \frac{1}{\Delta x^2}$
However, commons sense would suggest the idea that it might be possible to do better, and obtain a bounding factor of the form $ \frac{1}{\Delta x ^{\frac{3}{2}}} $, by not moving outside the norm $(I-Q)^{-1} $ and letting it take care (in some way) of $(I-Q)$ .
The question is, if this is actually possible, how can one prove it?