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For arbitrarily small values of $\varepsilon>0,$ we formulate and analyse theMaxwell system of equations of electromagnetism on $\varepsilon$-periodic sets$S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures$\mu^\varepsilon,$ such that ${\rm supp}(\mu^\varepsilon)=S^\varepsilon,$ isobtained by $\varepsilon$-contraction of a fixed 1-periodic measure $\mu,$ andfor right-hand sides $f^\varepsilon\in L^2({\mathbb R}^3, d\mu^\varepsilon),$we prove order-sharp norm-resolvent convergence estimates for the solutions ofthe system. Our analysis includes the case of periodic "singular structures",when $\mu$ is supported by lower-dimensional manifolds. The estimates areobtained by combining several new tools we develop for analysing the Floquetdecomposition of an elliptic differential operator on functions from Sobolevspaces with respect to a periodic Borel measure. These tools include ageneralisation of the classical Helmholtz decomposition for $L^2$ functions, anassociated Poincar\'{e}-type inequality, uniform with respect to the parameterof the Floquet decomposition, and an appropriate asymptotic expansion inspiredby the classical power series. Our technique does not involve any spectralanalysis and does not rely on the existing approaches, such as Bloch wavehomogenisation or the spectral germ method.

Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures