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This work is devoted to the development and analysis of a linearizationalgorithm for microscopic elliptic equations, with scaled degenerateproduction, posed in a perforated medium and constrained by the homogeneousNeumann-Dirichlet boundary conditions. This technique plays two roles: toguarantee the unique weak solvability of the microscopic problem and to providea fine approximation in the macroscopic setting. The scheme systematicallyrelies on the choice of a stabilization parameter in such a way as to guaranteethe strong convergence in $H^1$ norm for both the microscopic and macroscopicproblems. In the standard variational setting, we prove the $H^1$-typecontraction at the micro-scale based on the energy method. Meanwhile, we adoptthe classical homogenization result in line with corrector estimate to show theconvergence of the scheme at the macro-scale. In the numerical section, we usethe standard finite element method to assess the efficiency and convergence ofour proposed algorithm.

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production