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.

这项工作致力于开发和分析微观椭圆方程的线性化算法，并按比例缩小简并产量，并将其置于多孔介质中，并受齐次Neumann–Dirichlet边界条件的约束。该技术扮演两个角色：确保微观问题的独特弱可解性，并在宏观环境中提供良好的近似。该方案系统地依赖于稳定参数的选择，以确保无论是微观还是宏观问题，\（H ^ 1 \）范式都具有很强的收敛性。在标准变量设置中，我们证明\（H ^ 1 \）基于能量方法的微尺度收缩。同时，我们采用符合校正器估计的经典均匀化结果，以显示该方案在宏观尺度上的收敛性。在数值部分，我们使用标准有限元方法来评估所提出算法的效率和收敛性。