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

## TL;DR

This paper introduces a linearization algorithm for semi-linear elliptic equations with variable scaled production in perforated media, ensuring strong convergence and solvability at micro and macro scales, validated through finite element simulations.

## Contribution

The work develops a novel linearization scheme that guarantees strong $H^1$ convergence for complex elliptic problems with degenerate production, combining micro-scale contraction and macro-scale homogenization.

## Key findings

- Proves $H^1$-contraction at the micro-scale using energy methods.
- Establishes convergence at the macro-scale via homogenization and corrector estimates.
- Numerical results confirm the efficiency and convergence of the proposed algorithm.

## Abstract

This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $H^1$ norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the $H^1$-type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.