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arXiv · 1905.07122

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

Anh-Khoa Vo·Ekeoma Rowland Ijioma·Nhu-Ngoc Nguyen
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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.

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Anh-Khoa Vo, Ekeoma Rowland Ijioma, Nhu-Ngoc Nguyen. 2019-05-17. Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production. https://arxiv.org/abs/1905.07122

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