A note on a posteriori error bounds for numerical solutions of elliptic equations with a piecewise constant reaction coefficient having large jumps

We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation $\Delta \Delta u + {{\Bbbk }^{2}}u = f$, where the coefficient $\Bbbk \geqslant 0$ is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to $\Bbbk \in [0,{\text{c}}{{{\text{h}}}^{{ - 2}}}]$, $c={\text{const}}$ , and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for $\Bbbk \equiv {\text{const}}{\text{.}}$ The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017–2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.