# Quantification of non-stationarity of PDE solution

Suppose I have a time-dependent PDE discretized by the Rothe method and FEM, like

$$\int_{\Omega} k^{n+1/2}(u^{n+1}-u^{n}) v \;\mathrm{d}x = F^{n+1/2}(u^{n+1},u^n)[v] \quad \forall v\in V_h^n.$$

What is a good way of quantifying the difference between a transient solution $u^{n} \in V_{h}^{n}$ and a stationary solution $u^{n}_{\mathrm{S}}$ (satisfying the given equation with $k^{n+1/2} = 0$), without actually computing $u_{\mathrm{S}}$ and $\|u^{n} - u^{n}_{\mathrm{S}}\|$?

Note that $F^n$, $k^{n+1/2}$, and $V^n_h$ all depend on $n$ so that stationary solutions $u^n_\mathrm{S}$ are different at each time level $t_n$.

I'm targeting the computation of norms of some element tensors etc.

You could look at the residual: $\|F(u^n)\|$. If $u^n=u_S$ then $F(u_S)=0$, but this isn't true for a transient solution.

Let's assume that stationary solutions $u^n_\mathrm{S}$ are given by $$0 = F^{n+1/2}(u^{n+1}_\mathrm{S}, u^n_\mathrm{S}) [v] .$$ We can do that - imagine that $F^{n+1/2}$ is Crank-Nicolson discretization; then this means $$0=F^{n+1/2}(u^{n+1/2}_\mathrm{S}) [v] .$$

Let's define $$r^n := u^n - u^n_\mathrm{S} .$$

Now we use Taylor theorem on the RHS of the first equation (assuming $r^n$, $r^{n+1}$ are small) and we have up to the first order $$0 \approx F^{n+1/2}(u^{n+1}, u^n) [v] + D_{u^{n+1}}F^{n+1/2}(u^{n+1},u^n) [v][-r^{n+1}] + D_{u^{n }}F^{n+1/2}(u^{n+1},u^n) [v][-r^{n }] .$$

Now define notation $R^{n+1/2} := F^{n+1/2}(u^{n+1}, u^n)$, $J^{n+1/2} := D_{u^{n+1}}F^{n+1/2}(u^{n+1},u^n)$ and $\widetilde{J}^{n+1/2} := D_{u^{n }}F^{n+1/2}(u^{n+1},u^n)$ so that last approximation reads $$0 = R^{n+1/2} [v] - J^{n+1/2} [v][r^{n+1}] - \widetilde{J}^{n+1/2} [v][r^{n }] .$$

Picking $v:=r^{n+1}$ and rearranging last equation we have $$J^{n+1/2}[r^{n+1}][r^{n+1}] = R^{n+1/2} [r^{n+1}] - \widetilde{J}^{n+1/2} [r^{n+1}][r^{n }] .$$

If $J^{n+1/2}$ is elliptic $J^{n+1/2}[v][v]\geq\alpha||v||^2$ $$||r^{n+1}|| \leq \frac{1}{\alpha}\left|\left|R^{n+1/2} - \widetilde{J}^{n+1/2} [\cdot][r^{n }]\right|\right| ,$$ which could be useful as reccursion relation for $||r^n||$. In a special case where $F^{n+1/2}$ is backward Euler discretization so that $\widetilde{J}^{n+1/2} = 0$ one have simply $$||r^{n+1}|| \leq \frac{1}{\alpha}\left|\left|R^{n+1/2}\right|\right| .$$ LHS of the last equation is target quantity and norm on the RHS can be calculated cheaply from already computed transient solutions. The biggest trouble is $\alpha$, the ellipticity constant of the Jacobian $J^{n+1/2}$